Lecture 1a. DESCRIPTIVE AND INFERENTIAL STATISTICS
 Experimentalists use randomized experiments.
 Differentialist uses observational studies and correlations.
 Randomized experiments
 random and representative sampling
 independent and dependent variables
 doubleblind experiments
 confounds
 causality
 Causality: Randomized controlled experiments allow
for strong claims about causality. You can predict,
prevent bad, promote good things.
 Strong causal claims require
 true independent variables
 random and representative samples
 no confounds
 Population: the entire collection of cases we want to
generalize (e.g. all children in the U.S.).
 Sample: a subset of the population.
 Parameter: Numerical measurement that describes a
population characteristic.
 Statistic: numerical measurement that describes a
characteristic of a sample.
 Descriptive statistics: procedures to summarize,
organize and simplify data.
 Inferential Statistics: technique to generalize about
population parameters based on sample statistics.
 Independent variable: variable manipulated by the
experimenter.
 Dependent variable: variable affected by the independent
variable.
Lecture 1b. OBSERVATIONAL STUDIES.
 Study vs. experiment.
 Patterns of correlations.
 Quasiindependent variables:
 Intelligence testing
 Intelligence models have been proposed
based on correlations across different
tests.
 These are studies as no variable is
being manipulated.
 No arguments can be made about causality.
 Effects of concussion
 Quasiindependent variables
 Treatment
 Suffered a sportsrelated concussion.
 Control group.
 Dependent variable
 Neural measures.
 Cognitive measures.
 Athletes aren't randomly assigned.
 Confounds
 Prior concussions.
 Prior hits to the head.
 Different personality types
(aggressive).
 Subtle personality types, characteristics
(ie.they might be really fast and get
into those situations more often.)
 These could lead to the outcomes seen in
the dependent variables.
 These are observational studies, not
randomized experiments. Therefore,
causality arguments are weak.
 Important concepts
 Study vs. experiment.
 Patterns of correlations.
 Quasiindependent vs independent
variables.
Lecture 2a. DESCRIPTIVE STATISTICS
 Histograms
 Summary statistics.
 Tools for inferential statistics.
 Histograms concepts.
 What is distribution.
 The normal distribution.
 Many nonnormal distributions
 Rather than average or summarize information, show the
entire distribution.
 A bias is a difference in measurements (i.e.
thermometer wand measures higher than the glass
thermometer).
 Histograms often revel important information.
 Sex differences in spatial reasoning.
 In some statistics the mean is higher in men,
BUT
 The variance in each group is far greater than the gap
between groups.
 Observing the histograms, you see the difference was in
the peaks. Men had a flatter distribution. More woman
scored in the mean than men.
 The pill: The mean suggested the pill causes slightly
higher blood pressure in the mean for women who took
the pill.
 The histogram showed it raised the blood pressure for all
women. The whole graph was moved 4 units to the
right.
 Those histograms also showed 2 peaks. One for smokers
and one for nonsmokers.
 Examples
 Uniform: flat, not a bell curve. Data was uniformly spaced.
 Positively skewed: the positive end is flatter, or has skew.
 Negatively skewed: the negative end if flatter, or has skew.
 Skew is where there are few in the distribution.
 Summarization of a distribution
 Central tendency.
 Variability.
 Skew.
 Kurtosis (a peak or flat curve).
 In mathematics these are known as the four "moments"
of the mean.
 Important concepts.
 What is a distribution.
 The normal distribution.
 Many nonnormal distributions.
Lecture 2b. SUMMARY STATISTICS
 Summary statistics.
 Central tendency (mean, median, mode).
 Variability (standard deviation and variance).
 Skew.
 Kurtosis.
 Some people don't want the entire distribution. Marketers want
to know what product sells best, the mean. They want central
tendency.
 Measures of central tendency.
 Measurement of the center point of distribution. It must be
representative.
 Mean: average M = ∑(X) / N.
 Median: middle score.
 Mode: most common score.
 Mean is used when the distribution is normal.
 Median is preferred when there are extreme scores in the
distribution.
 Mode is the score that occurs most often, the peak of a
histogram.
 Variability: range and diversity of a distribution  width.
 Standard deviation: average deviation from the mean.
 Variance: standard deviation squared.
 Deviations sum to zero. If you square them
the negative sign drops out.
 M = mean = average = ∑(X) / N.
 X  M = deviation.
 ∑(X  M)² = sum of deviations²
= sum of squares = SS.
 ∑(X  M)² / N = variance = mean squared
= MS.
 √(MS) = standard deviation = SD.
 SUMMARY STATISTICS
 Central tendency (mean, medium, mode).
 Variability (standard deviation, variance).
 Skew.
 Kurtosis.
 SD^{2} = ∑( X  M )^{2} / N.
Lecture 2c. TOOLS FOR INFERENTIAL STATISTICS
 Tools for inferential statistics:
 Normal distribution.
 Zscores.
 Percentile rank.
 Probability.
 Inferential statistics.
 Normal distribution
 Bell shaped and symmetrical around the center point.
 The thermometer wand measured hot. That is a bias.
 Zscores
 Z = ( X  M ) / SD.
 Z of the mean = 0.
 Z of the standard deviation = 1.
 Percentile rank: percentage of scores that fall at or
below a given score.
 Percentile rank of the mean = 50%
 i.e. Z of +1, if 34% of the scores are between
the mean and a Zscore of + 1, 50% + 34% = 84%.
 You should be able to switch between
 Raw scored
 Zscored
 Percentile rank
with relative ease.
 Probability (of an event)
 P(E) = (# of ways E can be attained) / (# of possible
outcomes).
 Probability and normal distribution.
 Inferential statistics
 Assume a normal distribution.
 Assume certain values, such as a mean.
 Conduct an experiment.
 Do the assumption hold. Either way, an inference
can be made.
 Is is safe to assume a normal distribution.
 What are you trying to measure.
 What is the construct.
 How do you operationalize the construct.
 Tools review
 Normal distribution.
 Zscores.
 Percentile ranks.
 Probability.
 Inferential statistics.
Lecture 3a. INTRODUCTION TO R
 Why R
 It's free.
 Works on any platform.
 Opensource.
 Flexible.
 Excellent graphics.
 Widely used in academia and business.
 To use R in linux (Debian/Ubuntu/Kubuntu)
 sudo aptget install rbase rcranboot
rcranclass rcrancluster rcranvgam
 Create scripts in a text editor, save as
.R files.
 # is a comment.
 From the command prompt
type "R {Enter}".
 Test: type 5+3 {Enter}
 The directory you are in is the
working directory.
 The professor wants us to use the
levene.test() in install.packages("car").
This is linux. That one only returns
the median. The professor asks for the
mean. To get the mean, use
install.packages("lawstat").
 Before you can do that you have to
"sudo aptget install
rcranvgam".
 install.packages("psych")
 library(psych)
 search()
 list.files()
 source( "file.name.R", echo=T )
 help( function.name )
 quit()
 For Windows: Download and install R
 http://www.rproject.organize
 Click on the CRAN link.
 Choose a country and city.
 Select your operating system.
 From the R console types
 >install.packages("psych")
>library(psych) # load the package
# denotes a comment.
>search() # lists loaded packages.
 "File" menu
 "New Document".
 R editor appears.
 Write a line of code.
 Press return or enter.
 Result is returned.
 For example, use R as a calculator:
 R editor
 Write several lines of code.
 Save as a "script".
 Execute script.
 Results are returned in the R console.
 Set you preferences
 Establish a "working directory"
 Click on R.
 Click on "Preferences".
 Click on "Startup".
 Click on "Change" directory.
 Choose a "Path" ( i.e
~/Common/Language/R )
Lecture 3b. WRITING A SCRIPT
 Script
 Several lines of code.
 Composed in the R editor.
 Comment and save for later.
 Goals for this script
 Read data into a data.frame.
 Explore the contents of the data.frame.
 Plot histograms.
 Get descriptive statistics.
 On the course web sight, get and use the text file
 First line(s) of code should be comments.
 # Statistics One, Lecture 3, example script.
 # Read data, plot histograms, get descriptives.
 Read data into a data.frame.
 We will name the data.frame "ratings".
 ratings < read.table("stats1.ex.01.txt",
header = T)
 Explore the contents of the data.frame.
 class(ratings)
 R will return "data.frame"
 name(ratings)
 R will return "RedTruck" "WoopWoop"
"HobNob" "RedTruck"
 Plot histograms
 hist(RedTruck)
 R returns a histogram for RedTruck.
 Plot four histograms on one page.
 layout( matrix( c( 1,2,3,4 ), 2, 2, byrow = TRUE ) )
 hist( ratings$WoopWoop, xlab = "Rating" )
 hist( ratings$RedTruck, xlab = "Rating" )
 hist( ratings$HobNob, xlab = "Rating" )
 hist( ratings$FourPlay, xlab = "Rating" )
 R returns 4 histograms on one page.
 Explain code
 layout( matrix( c
 c( 1,2,3,4 ), 2, 2
 histograms for items 1  4, in a 2 x 2 matrix, by row.
 xlab: x axis label = Rating.
 help(hist) gives all the options available for
hist.
 $ pick out one variable in ratings data.frame.
 Get descriptive statistics
 Functions used"
 read.table
 class
 names
 hist
 describe
 Learn more about functions
 In R types
 >help(hist)
 >help(read.table)
 Or google "how to do something in R".
 Final Product
 Script
 Histograms
 Descriptive statistics
COMPLETE SAMPLE SCRIPT
# Statistics One, Lecture 3, example script
# Read data, plot histograms, get descriptives
install.packages("psych")
library( psych )
#Verify
search()
# Read the data into a data.frame called ratings
ratings < read.table("~/Common/Language/R/stats1.ex.01.txt",
header = T)
# What type of object is ratings?
class( ratings )
# List the names of the variables in the data.frame
called ratings
names( ratings )
# Print 4 histograms on one page
layout( matrix( c( 1,2,3,4 ), 2, 2,
byrow = TRUE ) )
# Plot histograms
hist( ratings$WoopWoop, xlab = "rating" )
hist( ratings$RedTruck, xlab = "rating" )
hist( ratings$HobNob, xlab = "rating" )
hist( ratings$FourPlay, xlab = "rating" )
# Descriptive statistics for the variables in the
data.frame called ratings
describe( ratings )
 To run a file ending in .R
 list.files()
 source( "filename.R", echo = T )
ASSIGNMENT ONE CODE
# Statistics One, Week One, Assignment 1
# Designed sports training on working memory study
# Read data, plot histograms, get descriptives
library(psych)
# Read the data into a data.frame called training
training < read.table("~/Common/Language/R/daa.01.txt", header = T)
# Break training data into two subsets
training_des < subset( training, subset=( cond=="des" ) )
training_aer < subset( training, subset=( cond=="aer" ) )
# What type of object is training?
class( training )
# What type of object is training_des, training_aer?
class( training_des )
class( training_aer )
# List the names of the variables in the data.frame called training
names( training )
names( training_des )
names( training_aer )
# Print 8 histograms on one page
layout( matrix( c(1,2,3,4,5,6,7,8), 2, 4, byrow = TRUE ) )
hist( training_des$pre.wm.s )
hist( training_des$post.wm.s )
hist( training_des$pre.wm.v )
hist( training_des$post.wm.v )
hist( training_aer$pre.wm.s )
hist( training_aer$post.wm.s )
hist( training_aer$pre.wm.v )
hist( training_aer$post.wm.v )
# Descriptive statistics for the variables in the data.frame called training
describe( training_des )
describe( training_aer )
ASSIGNMENT ONE TEST
Question 1 
Which of the following best describes
the shape of the distribution of wm.s scores in the
Aerobic Condition before training? 
 Normal 
 Positively skewed 
X  Negatively skewed 
 uniform 
Question 2 
Which of the following best describes
the shape of the distribution of wm.v scores in the
Designed Sports conditions before training? 
 Multimodal 
 Uniform 
 Bimodal 
x  Normal 
Question 3 
Which variable revealed the greatest
variance before training? 
x  Aerobic  wm.v 
 Designed Sports – wm.v 
 Designed Sports  wm.s 
 Aerobic  wm.s 
Question 4 
Which variable appears to be most affected
by training? 
x  Aerobic  wm.s 
 Aerobic  wm.v 
 Designed Sports  wm.s 
 Designed Sports  wm.v 
Question 5 
More analyses are necessary but at first
glance does it appear that the data support the hypothesis
that designed sports training improves spatial WM to a
greater extent than it improves verbal WM? 
x  True 
 False 
QUIZ ONE
Question 1 
A fundamental flaw in the memory training
experiment discussed in lecture was: 
 The memory task was unrelated to the intelligence test 
 There wasn't random assignment to conditions 
 The effect of memory training wasn't related to the amount of training 
x  The control condition was too different from the treatment condition 
Question 2 
The major benefit of randomized controlled experiments
is that they allow for strong arguments about: 
x  Causal relationships between independent and dependent variables 
 Shapes of distributions 
 Differences between men and women 
 The difference between histograms and density plots 
Question 3 
The distribution of household income in the United States,
currently, is: 
 Negatively skewed 
 Uniform 
 Normal 
x  Positively skewed 
Question 4 
The study discussed in lecture about the effect of
contraception on blood pressure was somewhat convincing, despite
not being a randomized controlled experiment, because: 
x  The effect was consistent across the entire range of the distributions 
 The sample was large 
 There was random selection from the population 
 The distributions in each group were relatively normal 
Question 5 
Two independent groups of people may differ in the
mean (average) on some variable. However, they may also differ on: 
 Kurtosis 
 Variance 
 Skew 
x  Variance, Skew and Kurtosis 
Question 6 
When distributions are skewed, the most
accurate measure of central tendency is: 
 The skew 
x  The median 
 The kurtosis 
 The mean 
Question 7 
Given a distribution of scores, the
average of the squared deviation scores is equal to: 
x  The variance 
 The standard error 
 The sum of squares 
 The standard deviation 
Question 8 
Suppose that the “wand” (infrared meter)
measure of body temperature systematically overestimates
temperature by 1 degree Fahrenheit. This would be an example
of: 
 A confound 
x  Measurement bias 
 Standard error 
 Chance error 
Question 9 
If X = 200, M = 300, SD = 100, then Z = 
 0.5 
 1 
 0 
x  1 
Question 10 
If the Sum of Squared deviation scores
(SS) = 100 and there are 25 people in the sample then
the Standard Deviation (SD) is: 
x  2 
 1 
 4 
 0 
Lecture 4a: CORRELATIONS
 Correlation
 A statistical procedure used to measure and describe
the relationships between two variables.
 Correlations can range between +1 and 1.
 +1 is a perfect positive correlation.
 1 is a perfect negative correlations.
 0 is no relationship between two variables
at all.
 Example: working memory capacity (X) is strongly correlated
with SAT score (Y). If I know a persons score on a working
memory task then I can predict their SAT score.
 R scatterplot: plot( SAT ~ WMC )
 CAUTION: Accuracy of predictions will depend upon:
 The magnitude of the correlation.
 The magnitude of the correlation depends upon:
 Reliability of X.
 Reliability of Y.
 Sampling (random and representative).
 The validity of the prediction depends upon:
 Validity of X.
 Validity of Y.
 Several other assumptions (in Segment 3).
 Most important:
 The correlation coefficient is a sample statistic.
It does not apply to individual cases in the
sample.
 A more serious example: Intelligence testing and WWI.
 The advent of WWI caused a unique challenge for the U.S. military.
 How to quickly recruit and assign to positions
a large number of new men.
 Specifically, which men should be designated as
officers and/or assigned to officer training.
 Historical coincidence
 Intelligence testing was also new. It used a
scientific approach to individual differences in
intelligence.
 The Army Alpha Battery
 Prominent Psychology researchers were recruited by
the military to develop an aptitude test that could
be administered to large groups of men and quickly
scored.
 One of them, Robert Yerkes, argued that a specific
test measured "native intellectual ability" and
was unaffected by culture.
 However, later research demonstrated that it was
clearly culturally biased.
 What type of statistical information could be
presented to support Yerkes' claim?
 What type of statistical information could be
presented to refute Yerkes' claim?
 The decisive answer was that the test held for one
culture group, but not others.
 Baseball and Sabermetrics
 Which statistic is a better predictor of a player's
contribution to the team's offense, AVG or OBP.
 In data collected from the New York Yankees:
 R scatterplot: plot(
R ~ AVG )
 R scatterplot: plot(
R ~ OBP )
 Important Concepts
 What is a correlation?
 What are they used for?
 Scatterplots.
Lecture 4b: CALCULATING CORRELATIONS
 Important topics:
 Pearson product moment correlation coefficient (r).
 Covariance
 Correlation coefficient (r)
 r = the degree to which X and Y vary together,
relative to the degree to which X and Y vary
independently.
r = 
covariance of X and
Y 
variance of X and Y 
 Formulas for r:
 Raw score formula.
 Zscore formula.
 Sum of cross products.
 Review calculation for SS (sum of squares).
 SS_{x} = ∑[ ( X  M_{x}
)^{2} ]
 To calculate SP (sum of cross products )
 For each subject in the sample, calculate
their deviation scores on both the X and Y.
 ( X  M_{x} )
 ( Y  M_{y} )
 For each subject multiply the deviation
score on X by the deviation score on Y:
 ( X  M_{x} ) ( Y  M_{y} )
 Then sum the "cross products"
 SP = ∑[ ( X  M_{x} ) ( Y 
M_{y} ) ]
 Formula to calculate r:
 Raw score formula:
r = 
SP_{xy}

√ (SS_{x}
SS_{y}) 
 SS_{x} = ∑( X  M_{x}
)^{2}
 SS_{y} = ∑[ ( Y  M_{y} )
( Y  M_{y} ) ]
 SP_{xy} = ∑[ ( X  M_{x} )
( Y  M_{y} ) ]
r = 
∑[
( X  M_{x} )
( Y  M_{y} )
] 
[ ∑( X  M_{x}
)^{2}
∑( Y  M_{y}
)^{2}
]^{½} 
 Zscore formula:

Z_{x} = 
( X  M_{x}
) 
SD_{x} 
Z_{y} = 
( Y  M_{y}
) 
SD_{y} 
SD_{x} = 
√∑(X
 M_{x})² 
N 
SD_{y} = 
√∑(X
 M_{y})² 
N 
 Variance and covariance:
 Variance = MS (mean^{2})
= SS / N
 Covariance = COV = SP / N
 Correlation is standardized COV
 Standardized so the value is in the range 1 to 1.
 Note on the denominators.
 Correlation for descriptive purposes:
 Correlation for inferential purposes:
Lecture 4c: INTERPRETATION OF CORRELATIONS
 Important Topics
 Validity of a correlationbased argument.
 Reliability of a correlation.
 The validity of any argument made on the basis
of a correlation analysis depends on these
assumptions:
 Normal distribution for X and Y.
 To detect for violations:
 Plot histograms and run descriptive
statistics.
 Linear relationship between X and Y (not
quadratic):
 To detect violations:
 Examine scatter plots.
 Plot a histogram of residuals.
 Homoskedasticity
 How to detect violations:
 Examine scatter plots.
 Plot a histogram of residuals.
 Homoskedasticity
 The standard deviation is consistent across the
regression line, homogeneity of variance.
 In a scatter plot the distance between a dot and
the regression line reflects the amount of
prediction error, or the residual.
 If it is homoskedastic, those distances are
not a function of X, not related to the
predictive variable. If they are you might
have a confound in you study.
 In a scatter plot, if the data points are
scattered randomly in relation to the
regression line it is homoskedastic.
If the variance changes with X, they are
heteroskedastic.
 Reliability of a correlation:
 Does the correlation reflect more than
just a chance of covariance?
 If the correlation is iffy, how do you determine
across samples if it's valid.
 One approach to this question is to use
NHST.
 If there is not a correlation, what are the odds
that you would get these results.
 NHST  Null hypothesis significance testing.
In R's cor.test(), this is the "pvalue" of
the "cor"..
 H_{0} = null hypothesis: r = 0.
 H_{A} = alternative hypothesis:
r > 0.
 Assume H_{0} is true, then calculate the
probability of observing data with these characteristics,
given H_{0} is true.
 Thus, p = P(DH_{0}) ( =
given)
 If p < α then reject H_{0},
else retain H_{0}
Truth  
Retain H_{0}  Reject
H_{0} 
H_{0} true 
Correct decision 
Type I error (false alarm) 
H_{0} false 
Type II error (miss) 
Correct decision 
Truth  
Retain H_{0}  Reject
H_{0} 
H_{0} true 
p = ( 1  α ) 
p = α 
H_{0} false 
p = β ( 1  POWER ) 
p = ( 1  β ) POWER 
 p = P( D  H_{0} )
 Given that the null hypothesis is true,
the probability of true, or more
extreme data, is p.
 NOT: The probability of the null
hypothesis.
 In other words, P( D  H_{0} )
<> P( H_{0}  D )
 NHST can be applied to:
 r: Is the correlation significantly
different from zero?
 r1 vs. r2: Is one correlation
significantly larger than
another?
 There are other correlation coefficients.
 Point biserial r:
 When one variable is continuous and
the other is dichotomous.
 Phi coefficient:
 When both variables are
dichotomous.
 Spearman rank correlation:
 When both variables are ordinal
(ranked data).
 Important topics:
 Validity of a correlationbased argument.
 Reliability of a correlation.
Lecture 5a: MEASUREMENT
 Reliability and Validity
 Classical test theory (true score theory):
 Raw score (X) are not perfect.
 They are influenced by bias and
chance error.
 In a perfect world, we would obtain
a "true score".
 X = true score + bias +
chance error
 A measure (X) is considered to be
reliable as it approaches the true
score.
 The problem is we don't know the
true score.
 So we estimate reliability.
 Reliability estimates and methods of testing.
 Test / retest
 Measure everyone twice.
 The correlation between them
is an estimate of
reliability.
 If a bias is uniform, we
won't detect it.
 Parallel tests:
 In the body temperature example,
besides the wand, also use an
oral thermometer.
 The correlation between the two
tests is still an estimate of
reliability but now the bias
will be revealed.
 Interitem estimates:
 The most commonly used method in social
sciences because it's cheapest.
 For example, suppose a 20item survey
is designed to measure extraversion.
 Randomly select 10 items to get
subset A (X1).
 The other 10 items become subset
B (X2).
 The correlation between them is an
estimate of reliability.
 Construct validity:
 What is a construct?
 An ideal "object" that is not directly
observable.
 As opposed to "real" observable
objects.
 For example, "intelligence" is a
construct.
 How do we operationalize a construct?
 The process of defining a construct to
make it observable and quantifiable
(i.e. intelligence
tests).
 Construct validity.
 Content validity.
 Convergent validity.
 Divergent validity.
 Nomological validity.
 Example construct: verbal ability in children.
 Operationalize it with a vocabulary test.
 Content validity: does the test consist
of words that children should know?
 Convergent validity: Does the test
correlate with other, established
measures of verbal ability? For
example, reading comprehension.
 Divergent validity: does the test correlate
less will with measures designed to test a
different type of ability? For example,
spatial reasoning.
 Nomological validity: are scores on the
test consistent with more general theories,
for example, of child development and
neuroscience? For example, children with
damage or disease to brain regions
associated with language development should
score lower on the test.
Lecture 5b: SAMPLING
 Sampling error:
 From the wine example, suppose there are only 300
certified "wine experts" in the entire world.
 The population N = 300.
 Using Red Truck, let's assume a normal
distribution in the population with
a mean of 5.5 and standard deviation
of 2.22. The histograms looks
different also.
 Now take a random sample, N = 30, with a mean
of 5.93 and standard deviation of 2.45.
 This is sampling error.
 Increase the sample to N = 100. Now
we get a mean of 5.47 and a standard
deviation of 2.19. The histogram look
right. This is much closer to the
population.
 Go the other way, a sample of N = 10.
Now we bet a mean of 6.00 and standard
deviation of 1.7. The histogram doesn't
look at all like the population
one.
 Sampling error is the difference between
the population and the sample.
 Typically, we don't know the population
parameter. So how do we estimate
sampling error?
 Clearly, sampling error depends on the
size of the sample, relative to the
population
 It also depends on the variance in
the population. A large variance
will lead to greater sampling
error.
 Standard error:
 The estimate of amount of sampling error:

 SE: Standard error.
 SD: standard deviation of the
sample.
 N: Size of the sample.
 Probability histograms:
 Standard error is the standard deviation
of the probability histogram.
 If a variable X is perfectly normal, we
know a lot about it's distribution. We
know what scores fall in what regions.
 It is symmetrical, so 50% of the
distribution falls below the mean,
50% above.
 68% of the distribution falls
within 1 standard deviation of
the mean.
 Previous histograms where plots of
individuals. Now we have to consider
a distribution of samples.
 Probability histogram:
 A distribution of sample means:
 Assume we took multiple samples of
the same size and them plotted all
the sample means.
 Suppose we sampled N = 10, multiple
times.
 This histogram would have a normal
curve but the width would
be large. Each of the individual
samples would contribute varying
standard deviations.
 Samples of N = 100 would have
less fluctuation.
 Standard Error is the distance of 1
standard deviation (Z of 1),
in the distribution of sample
means.
 Distribution of sample means.
 The characteristics of this sample
means are hypothetical. We don't know
the dimensions of the distribution as
we do with individual score. We must
estimate.
 We will assume these characteristics:
 The mean of the distribution of
sample means = mean of the
population.
 The variance of the distribution of
sample means is less than the
variance in the population of
individuals

 the variance in the distribution
of the sample means =
the variance in the population
÷ by the size of the
sample.
 This formula is familiar. So remember,
these all have different meanings.
 σ^{2} (sigma²):
population variance.
 SD^{2}: variance in
the sample.
 SE (standard error):
standard deviation of the
distribution of samples.
 The shape of the distribution of
sample means is approximately
normal (if the samples are
large enough).
 Distribution of sample means:
 σ^{2}_{M}
is the variance of the
distribution of sample
means.
 σ_{M} is the
standard deviation of the
distribution of sample
means (standard error)
 σ^{2} is the
variance of the population.
 σ is the standard deviation
of the population.
 N is the sample size.
 Central limit theorem:
 The mean of the distribution of sample means
is the same as the mean of the population.
 The standard deviation of the distribution
of sample means is the square root of the
variance of the distribution of sample
means, which is
σ^{2}_{M} =
σ^{2} / N.
 Which means SE
= SD /
√N.
 (standard error) =
(standard deviation) /
(square root of the
size of the sample).
 The shape of the distribution of sample
means is approximately normal if either
(a) N >= 30 or (b) the shape of the
population distribution is normal.
Lecture 6a: CORRELATIONS IN R
 Goal of this script in R
 Histograms
 Descriptive statistics
 Scatterplots
 Correlations
 Data from ImPACT.com
 A computerized neuropsychological assessment of
memory and attention.
 Used to assess the cognitive effects of head
trauma, for example, sportsrelated
concussion.
 Impact main measures
 Verbal memory.
 Visual memory.
 Visual motor speed.
 Reaction time.
 Impulse control.
 Data available in
"stats1.ex.02.txt"
 Write a script
# Statistics One, Lecture 6a, example script
# Read data, plot histograms, get descriptives,
# examine scatter plots, run correlations.
library(psych)
# Read data into a data.frame called "impact"
impact < read.table("stats1.ex.02.txt", header=T)
# Explore data.frame
class(impact)
names(impact)
# Change default settings for graphics
par( cex = 2, lwd = 2, col.axis = 200, col.lab = 200, col.main = 200,
col.sub = 200, fg = 200 )
# Print 5 histograms on one page
layout(matrix(c(1,2,3,4,5,6),2,3, byrow=TRUE))
hist(impact$memory.visual, xlab="Visual memory",
main="Histogram", col="red")
hist(impact$memory.verbal, xlab="Verbal memory",
main="Histogram", col="green")
hist(impact$speed.vismotor, xlab="Visualmotor speed",
main="Histogram", col="blue")
hist(impact$speed.general, xlab="General speed",
main="Histogram", col="black")
hist(impact$impulse.control, xlab="Impulse control",
main="Histogram", col="purple")
describe(impact)
# Scatterplots
plot(impact$memory.verbal ~ impact$memory.visual,
main="Scatterplot",ylab="Verbal memory",
xlab="Visual Memory")
abline(lm(impact$memory.verbal ~ impact$memory.visual),
col="blue")
# Correlations (one pair at a times)
cor(impact$memory.verbal, impact$memory.visual)
# Again, only add test for significance
cor.test(impact$memory.verbal, impact$memory.visual)
# Correlations (all in a matrix)
cor(impact)
# Install and load new package
install.packages("gclus")
library(gclus)
# Verify
search()
# Scatterplot matrix
library(gclus)
impact.r = abs(cor(impact))
impact.col = dmat.color(impact.r)
impact.o < order.single(impact.r)
cpairs(impact, impact.o, panel.colors=impact.col,
gap=.5, main="Variables Ordered and Colored
by Correlation")
Lecture 6b: TEST / RETEST
RELIABILITY ANALYSIS IN R.
 Assume 40 athletes took the ImPACT test twice.
Tests A and B, taken a couple months apart.
 These are normal athletes, not involved in
any head injury. See if the scores between
the two tests correlate.
 How to structure repeated measures?
 Typical to add new columns.
 In R, sometimes it is better to add new rows.
We will demonstrate both.
# Statistics One,
Lecture 6b
# Test / retest reliability analysis,
column format
library( psych )
# Read data into a data.frame called impact.col
impact.col < read.table(
"stats1.ex.03.col.txt", header = T )
# List name of the variables in the data.frame
called impact.col
names( impact.col )
search()
describe( impact.col )
# Correlation ( A & B )
cor( impact.col$memory.verbal.A,
impact.col$memory.verbal.B )
cor( impact.col$memory.visual.A,
impact.col$memory.visual.B )
cor( impact.col$speed.vismotor.A,
impact.col$speed.vismotor.B )
cor( impact.col$speed.general.A,
impact.col$speed.general.B )
cor( impact.col$impulse.control.A,
impact.col$impulse.control.B )
# Statistics One,
Lecture 6b, example script
# Test / retest reliability analysis,
row format
library( psych )
# Read data into a data.frame
called impact.row
impact.row < read.table(
"stats1.ex.03.row.txt",
header = T )
# List name of the variables in the data.frame
called impact.row
names( impact.row )
search()
describeBy( impact.row, impact.row$test )
# Correlation ( A & B )
cor( impact.row$memory.verbal[ impact.row$test=="A" ],
impact.row$memory.verbal[ impact.row$test=="B" ] )
cor( impact.row$memory.visual[ impact.row$test=="A" ],
impact.row$memory.visual[ impact.row$test=="B" ] )
cor( impact.row$speed.vismotor[ impact.row$test=="A" ],
impact.row$speed.vismotor[ impact.row$test=="B" ] )
cor( impact.row$speed.general[ impact.row$test=="A" ],
impact.row$speed.general[ impact.row$test=="B" ] )
cor( impact.row$impulse.control[ impact.row$test=="A" ],
impact.row$impulse.control[ impact.row$test=="B" ] )
QUIZ TWO
Grade: 9 out
of 10 
Question
1 
When running correlation
analyses, how does one check to see if
the homoscedasticity assumption is
violated? 
 Examine the histogram of
X 
X  Examine the scatterplot
of X and Y 
 Examine the histogram
of Y 
 Conduct a regression
analysis 
Question
2 
Complete the following
syllogism: SS is to variance as SP is
to: 
X  Covariance 
 Standard error 
 Correlation 
 Standard deviation 
Question
3 
Variance is equal to the
_________________ of the of squared
deviation scores. 
 Correlation 
 Square root 
X  Sum 
 Mean 
Question
4 
In a scatterplot, the
distance between an individual dot and
the regression line represents: 
 Covariance 
X  Prediction Error 
 Homoscedasticity 
 Bias 
Question
5 
Pearson’s product moment
correlation coefficient (r) is used when
X and Y are: 
 Both dichotomous
variables 
 Both nominal
variables 
 Both categorical
variables 
X  Both continuous
variables 
Question
6 
If the correlation
between X and Y is r = 0 then for
any given score on X, the predicted
Y score will be: 
 The minimum score
on Y 
X  The average
of Y 
 The maximum score
on Y 
 The average o
f X 
Question
7 
We all know that
correlation does not imply causation
but correlations are useful because
they can be used to assess: 
 Validity 
 Reliability 
 Prediction
errors 
X  Reliability, Validity,
and Prediction errors 
Question
8 
Which of the
following pairs of variables is
most likely to be negatively
correlated? 
 SAT and college
GPA 
 SAT and IQ 
X  Hours watching TV
per week and college GPA 
 Hours studying per
week and college GPA 
Question
9 
Systematic
measurement error
represents: 
 chance
error 
 covariance 
 outliers 
X  bias 
Question
10 
Which of the
following is NOT a component of
construct validity: 
 convergent
validity 
X  sample
validity 
 divergent
validity 
 content
validity 
ASSIGNMENT TWO CODE
# Statistics One, Week Two Assignment
# Write R script to:
# Provide descriptive statistics for all 8
measures, for each condition.
# Provide an 8x8 correlation matrix for
each condition.
library(psych)
# Read the data into a data.frame called assing02
assing02 < read.table("~/Common/Language/R/daa.02.txt",
header = T, row.names = "pid" )
# row.names = "pid", removed the first column
from the data.frame.
# Break assing02 data into two subsets
assing02_des < subset( assing02, subset=(
cond=="des" ) )
assing02_aer < subset( assing02, subset=(
cond=="aer" ) )
# This is not necessary, but now break into
4 subsets
# "subset" works on rows. "select" works on
columns.
assing02_des_tst1 < subset( assing02_des,
select = c( pre.wm.s1, post.wm.s1, pre.wm.v1,
post.wm.v1 ) )
assing02_des_tst2 < subset( assing02_des,
select = c( pre.wm.s2, post.wm.s2, pre.wm.v2,
post.wm.v2 ) )
assing02_aer_tst1 < subset( assing02_aer,
select = c( pre.wm.s1, post.wm.s1, pre.wm.v1,
post.wm.v1 ) )
assing02_aer_tst2 < subset( assing02_aer,
select = c( pre.wm.s2, post.wm.s2, pre.wm.v2,
post.wm.v2 ) )
# What type of object is assing02?
class( assing02 )
# What type of objects are assing02_des,
assing02_aer?
class( assing02_des )
class( assing02_des_tst1 )
# List the names of the variables in
the data.frame called assing02
names( assing02 )
names( assing02_des )
names( assing02_des_tst1 )
# Descriptive statistics for the variables
in the data.frame called training
describe( assing02_des_tst1 )
describe( assing02_des_tst2 )
describe( assing02_aer_tst1 )
describe( assing02_aer_tst2 )
# This is the same thing as the 4
lines of code directly above.
describe( assing02_des [2:9] )
describe( assing02_aer [2:9] )
cor( assing02_des 2:9 )
cor( assing02_aer 2:9 )
ASSIGNMENT TWO TEST
Grade: 4 out of 5 
Question 1 
Which measures
displayed the lowest correlation
pretraining, suggesting the
weakest reliability? 
 Spatial working memory,
des condition 
 Verbal working memory,
aer condition 
X  Verbal working memory,
des condition 
 Spatial working memory,
aer condition 
Question
2 
Which measures
displayed the highest correlation
pretraining, suggesting the
strongest reliability? 
 Spatial working memory,
des condition 
 Verbal working memory,
aer condition 
X  Verbal working memory,
des condition 
 Spatial working memory,
aer condition 
Question
3 
In the aer
condition, which individual
measure displayed the highest
correlation between pre and post
training? 
X  wm.v2 
 wm.v1 
 wm.s2 
 wm.s1 
Question 4 
In the des
condition, which individual
measure displayed the highest
correlation between pre and
post training? 
 wm.v1 
X  wm.v2 
 wm.s2 
 wm.s1 
Question
5 
Based on the
correlations, the construct
to be interpreted with most
caution, from a measurement
perspective, is: 
 Verbal working
memory, des condition 
 Spatial working
memory, des condition 
 Verbal working
memory, aer condition 
X  Spatial working
memory, aer condition 
Lecture 7a: INTRODUCTION TO REGRESSION
 Regression equation and "model".
 A statistical analysis used to predict scores on an
outcome variable, based on scores on one or more
predictor variables.
 For example, we can predict how many runs a baseball
will score (Y) if we know the players batting
average (X).
 Y = m + bX + e
 Y is a linear function of X.
 m = intercept.
 b = slope.
 e = prediction error.
 Ŷ = B_{0} + B_{1}X_{1}
 Ŷ = predicted score.
 Y  Ŷ = e (prediction error).
 The regression model is used to "model" or predict
future behavior.
 The "model" is just an equation.
 The goal is to produce better models so we can
generate more accurate predictions.
 Add more predictor variables.
 Develop better predictor variables.
 In the baseball example, OBP predictions
were an improvement over AVG.
 Examine why the results got better.
 Examine residuals.
 Plot histogram.
 Scatterplot residuals.
 Linear relationship between X and Y.
 Homoscedasticity.
 Ordinary least squares estimation.
 The values of the coefficients (B) are
estimated such that the model yields
optimal predictions.
 Minimize the residuals!
 The sum of the squared (SS) residuals
is minimized.
 SS.RESIDUAL = ∑( Ý 
Y )^{2}.
 ORDINARY LEAST SQUARES estimation.
 Sum of Squared deviation scores (SS) in variable Y
 Sum of Cross Products (SP.XY)
 SS.X
 SS.Y
 Also called correlation or covariance.
 Sum of Squared deviation scores (SS) in variable Y.
 SS.Y = SS.MODEL + SS.RESIDUAL
 Unstandardized regression coefficients.
 How to calculate B (unstandardized)
 B = r ( SD_{y} /
SD_{x} )
 Standardized regression.
 Standardized regression coefficient =
β + r
 If X and Y are standardized then:
(one predictor)
 SD_{y} = SD_{x} = 1
 B = r ( SD_{y} /
SD_{x} )
 β = r
 This is only true in simple regression. Multiple
regression gets worse.
 Estimation of coefficients:
 Using the linear model function in R:
lm( Runs ~ OBP )
 Ŷ = B_{0} +
B_{1}X_{1}
 Ŷ = 282 + (1044)X
 Let X = 0.35
 Ŷ = 83
Lecture 7b: NHST: A closer look (null hypothesis
significance testing)
 Logic of NHST
 First, this probability is the pvalue returned
by R.
 H_{0} = null hypothesis: e.g., r = 0
 H_{A} = alternative hypothesis: e.g.,
r != 0
 From the regression standpoint, the unstandardized regression
coefficient, B in the regression equation.
 The slope relating X to Y is zero.
 H_{0} = null hypothesis: e.g., B = 0.
 H_{A} = alternative hypothesis: e.g., B != 0.
 Assume H_{0} is true, then calculate the probability of
observing data with these characteristics, given the
H_{0} is true:
 Thus, p = P(DH_{0})
 (p is a conditional probability, the probability of
obtaining this outcome (these data) given the assumption
that the null hypothesis is true.)
 If p < α, then Reject H_{0},
else Retain H_{0}.
Truth  
Retain H_{0}  Reject
H_{0} 
H_{0} true 
Correct decision 
Type I error (false alarm) 
H_{0} false 
Type II error (miss) 
Correct decision 
Truth  
Retain H_{0}  Reject
H_{0} 
H_{0} true 
p = ( 1  α ) 
p = α 
H_{0} false 
p = β ( 1  POWER ) 
p = ( 1  β ) POWER 
 p = P( D  H_{0} )
 Given that the null hypothesis is true,
the probability of true, or more
extreme data, is p.
 NOT: The probability of the null
hypothesis.
 In other words, P( D  H_{0} )
<> P( H_{0}  D )
 NHST can be applied to:
 r  Is the correlation significantly
different from zero?
 B  Is the slope of the regression line
For X significantly different from zero?
 NHST for B
 t = B / SE
 B is the unstandardized regression coefficient.
 SE = standard errors
 SE = √ SS.RESIDUAL / (N  2)
 It's good if t is close to zero.
 NHST problems
 Biased by N:
 pvalue is based on tvalue
 t = B / SE
 SE = √ SS.RESIDUAL / ( N  2 )
 Binary outcome:
 Technically speaking, one must Reject or Retain
the Null Hypothesis.
 What if p = .06?
 Null "model" is a weak hypothesis
 Demonstrating that your model does better than
NOTHING is not very impressive.
 NHST alternatives
 Effect size.
 Correlation coefficient (r).
 Standardized regression coefficient (B).
 Model R^{2}.
 Confidence intervals.
 Sample statistics are "point estimates"
Specific to the sample. They will vary
as a function of sampling error.
 Instead report "interval estimates".
The width of interval is a function of
standard error.
 Model comparison.
 Propose multiple models, Model A, Model B.
 Compare Model R^{2}.
Lecture 8a: MULTIPLE REGRESSION
 Intro to multiple regression
 Multiple regression equation
 Ŷ = B_{0} +
B_{1}X_{1}
+ B_{2}X_{2} +
B_{3}X_{3} ...
+ B_{k}X_{k}
 Ŷ = ∑(B_{0} +
B_{k}X_{k})
 Ŷ = predicted value on the
outcome variable Y.
 B_{0} = predicted value on
Y when all X = 0.
 X_{k} = predictor variables.
 B_{k} = unstandardized
regression coefficients.
 Y  Ŷ = residual (prediction
error).
 k = the number of predictor
variables.
 Interpretation of regression coefficients
 Model R and R^{2}
 R = multiple correlation coefficient.
 R = r_{ŶY}.
 The correlation between the
predicted scores and the
observed scores.
 R^{2}
 The percentage of variance in
Y explained by the model.
 Multiple regression Example
 Outcome measure (Y).
 Predictors (X1, X2, X3)
 Time since PhD (X1).
 # of publications (X2)..
 Gender (X3).
 Ŷ = 46, 911 + 1,382(time)
+ 502(pubs) + 3,484(G)
 Standard vs. sequential regression
 The difference between these
approaches is how they handle
the correlations among predictor
variables.
 if X1, X2, and X3 are not correlated
then type of regression analysis
doesn't matter.
 If predictors are correlated then
different methods will return
different results.
 STANDARD
 All predictors are entered into the
regression equation at the same
time.
 Each predictor is evaluated in terms
of what it adds to the prediction of
Y that is different from the
predictability offered by the
others.
 Overlapping areas are assigned to
R^{2} but not to any
individual B.
 SEQUENTIAL (aka hierarchical)
 Predictors are entered into the
regression equation in ordered
steps; the order is specified by
the researcher.
 Each predictor is assessed in terms of
what it adds to the equation at
its point of entry.
 Often useful to assess the change
in R^{2} from one step to
another.
Lecture 8b: MATRIX ALGEBRA
 A matrix is a rectangular table of known or
unknown numbers, e.g.
M = 
⌈ 1 2 ⌉  5 1   3
4  ⌊ 4 2 ⌋ 
 The size, or order, is given by
identifying the number of rows and
columns, e.g. the order of matrix M
is 4x2
 The transpose of a matrix is formed
by rewriting its rows as columns.
M^{T} = 
⌈ 1 5 3 4 ⌉
⌊ 2 1 4 2 ⌋ 
 Two matrices may be added or subtracted only
if they are of the same order.
M + N = 
⌈ 1 2 ⌉  5 1   3
4  ⌊ 4 2 ⌋ 
+ 
⌈ 2 3 ⌉  4 5   1
2  ⌊ 3 1 ⌋ 
= 
⌈ 3 5 ⌉  9 6   4
6  ⌊ 7 3 ⌋ 
 Two matrices may be multiplied when the number of
columns in the first matrix is equal to the number
of rows in the second matrix. If s, then we say
they are conformable for matrix multiplication.
 R = M^{T} * N
R_{ij} = ∑ (M^{T}_{ik}
* N_{kj})
R = M^{T} * N = 
⌈ 1 5 3 4 ⌉
⌊ 2 1 4 2 ⌋ 
* 
⌈ 2 3 ⌉
 4 5 
 1 2 
⌊ 3 1 ⌋ 
= 
⌈ 37 38 ⌉
⌊ 18 21 ⌋ 
 1*2 + 5*4 + 3*1 + 4*3 = 37
 1*3 + 5*5 + 3*2 + 4*1 = 38
 ∑( R_{2} * C_{1} ) = 18
 ∑( R_{2} * C_{2} ) = 21
 Special type of matrices
 A square matrix has the same number of rows as
columns.
D = 
⌈ 17 14 5 ⌉
 13 25 7 
⌊ 18 32 9 ⌋ 
 A square symmetric matrix is such that D =
D^{T}, the entries above the diagonal
are symmetric with the entries below the
the diagonal (i.e. correlation matrix):
D = 
⌈ 17 13 18 ⌉
 13 25 32 
⌊ 18 32 9 ⌋ 
 Diagonal matrices are square matrices with
zeros in all offdiagonal cells:
D = 
⌈ 17 0 0
⌉
 0 25 0 
⌊ 0 0 9 ⌋

 The inverse of a matrix is similar to the
reciprocal of a scalar. e.g., the inverse
of 2 is ½ and their product = 1.
 Inverses only exist for square matrices
and not necessarily for all square
matrices.
 The identity matrix is an inverse such that
D * D^{1} = I
I = 
⌈ 1 0 0 ⌉
 0 1 0 
⌊ 0 0 1 ⌋

 The determinant of a matrix is a scalar derived
from operations on a square matrix, by taking the
product of the diagonals relative to the
offdiagonals. For example, for a 2x2 matrix A
the determinant is denoted as A and is
obtained as follows:
 A = a_{11} * a_{22}
 a_{12} * a_{21}
 A vector is a matrix with only one row or
one column.
 EXAMPLE: A data matrix to a correlation matrix in 10 steps.
 Raw data matrix
Subjects as rows, variables as columns.
n = 10 (rows), p = 3 (columns).
X_{np} = 
⌈ 3 2 3 ⌉
 3 2 3 
 2 4 4 
 4 3 4 
 4 4 3 
 5 4 3 
 2 5 4 
 3 3 2 
 5 3 4 
⌊ 3 5 4 ⌋

 Row vector of sums (totals)
T_{1p} = 
1_{1p} * X_{np} = 
[ 1 1 1 1 1 1 1 1 1 1 ] * 
⌈ 3 2 3 ⌉
 3 2 3 
 2 4 4 
 4 3 4 
 4 4 3 
 5 4 3 
 2 5 4 
 3 3 2 
 5 3 4 
⌊ 3 5 4 ⌋ 
= [ 34 35 34 ] 
 Row vector of means
M_{1p} = T_{1p} *
N^{1} = [ 34 35 34 ] *
10^{1} = [ 3.4 3.5 3.4 ]
 Matrix of means
M_{np} = 
1_{n1} * M_{1p} = 
⌈ 1 ⌉
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
⌊ 1 ⌋ 
* [ 3.4 3.5 3.4 ] = 
⌈ 3.4 3.5 3.4 ⌉
 3.4 3.5 3.4 
 3.4 3.5 3.4 
 3.4 3.5 3.4 
 3.4 3.5 3.4 
 3.4 3.5 3.4 
 3.4 3.5 3.4 
 3.4 3.5 3.4 
 3.4 3.5 3.4 
⌊ 3.4 3.5 3.4
⌋_{47} 
 Matrix of deviation scores
D_{np} = 
X_{np}  M_{np} = 
⌈ 3 2 3 ⌉
 3 2 3 
 2 4 4 
 4 3 4 
 4 4 3 
 5 4 3 
 2 5 4 
 3 3 2 
 5 3 4 
⌊ 3 5 4 ⌋ 
 
⌈ 3.4 3.5 3.4
 3.4 3.5 3.4
 3.4 3.5 3.4
 3.4 3.5 3.4
 3.4 3.5 3.4
 3.4 3.5 3.4
 3.4 3.5 3.4
 3.4 3.5 3.4
 3.4 3.5 3.4
⌊ 3.4 3.5 3.4

⌉      
  ⌋ 
= 
⌈ .4 1.5
.4
 .4 1.5
.4
 1.4 .5
.6
 .6 .5
.6
 .6
.5
.4
 1.6 .5
.4
 1.4 1.5
.6
 .4 .5
1.4
 1.6 .5
.6
⌊ .4 1.5
.6 
⌉      
  ⌋ 
 Sums of squares and Crossproducts matrix
S_{xx} = D_{pn} *
D_{np} = 
⌈ 
⌊ 
.4 1.5 .4 
.4 1.5 .4 
1.4 .5 .6 
.6 .5 .6 
.6 .5 .4 
1.6 .5 .4 
1.4 1.5 .6 
.4 .5 1.4 
1.6 .5 .6 
.4 1.5 .6 
⌉ 
⌋ 
* 
⌈    
    ⌊

.4 1.5
.4
.4 1.5
.4
1.4 .5
.6
.6 .5
.6
.6
.5
.4
1.6 .5
.4
1.4 1.5
.6
.4 .5
1.4
1.6 .5
.6
.4 1.5
.6 
⌉      
  ⌋ 
= 
⌈  ⌊ 
10.4 2.0 .6 
2.0 10.5 3.0 
.6 3.0 4.4 
⌉  ⌋ 
 Variance/covariance matrix.
C_{xx} = S_{xx} *
N^{1} = 
⌈  ⌊ 
10.4 2.0 .6 
2.0 10.5 3.0 
.6 3.0 4.4 
⌉  ⌋ 
* 10^{1} = 
⌈  ⌊ 
1.04 .20
.06 
.20 1.05 .30 
.06 .30 .44 
⌉  ⌋ 
 Diagonal matrix of standard deviations.
S_{xx} =
(Diag(C_{xx}))^{1/2} = 
⌈  ⌊ 
1.02 0 0 
0 1.02 0 
0 0 .66 
⌉  ⌋ 
 Correlation matrix
R_{xx} =
S_{xx}^{1} *
C_{xx} *
S_{xx}^{1} = 
⌈  ⌊ 
1.02^{1}
0 0 
0 1.02^{1}
0 
0 0
.66^{1} 
⌉  ⌋ 
* 
⌈  ⌊ 
1.04 .20
.06 
.20 1.05 .30 
.06 .30 .44 
⌉  ⌋ 
* 
⌈  ⌊ 
1.02^{1}
0 0 
0 1.02^{1}
0 
0 0
.66^{1} 
⌉  ⌋ 
= 
⌈  ⌊ 
1.00
.19 .09 
.19 1.00
.44 
.09 .44
1.00 
⌉  ⌋ 
Lecture 8c: ESTIMATION OF COEFFICIENTS
 Still ORDINARY LEAST SQUARES estimation.
 But we will use matrix algebra.
 The values of the coefficients (B) are estimated
such that the model yields optimal predictions
 Minimize the residuals!
 The sum of the squared (SS) residuals is
minimized.
 SS.RESIDUAL = ∑(Ŷ Y)^{2}
 Estimation of coefficients.
 Ŷ = B_{0} +
B_{1}X_{1}
 Y  Ŷ = e (e is the prediction error,
or residual.)
 Ŷ = BX
# In both matrix form and standardized.
 Ŷ is a [N x 1] vector
 B is a [k x 1] vector
 k = number of predictors.
 X is a [N x k] matrix.
 Ŷ = BX
 To solve for B
 Replace Ŷ with Y
 Conform for matrix multiplication:
 Now let's make X square and symmetric.
 To do this, premultiply both sides of the
equation by the transpose of X, X'.
 Y = XB becomes
 X'Y = X'XB
 Now to solve for B, eliminate X'X.
 Premultiply by the inverse,
(X'X)^{1}
 X'Y = X'XB becomes
 (X'X)^{1}X'Y =
(X'X)^{1}X'XB
 Note that (X'X)^{1}X'X = I
 I = identity matrix. Any matrix
times the identity matrix returns
that matrix.
 And IB = B
 Therefore, (X'X)^{1}X'Y = B
 B = (X'X)^{1}X'Y
 Example
 Take the Sums of squares and Crossproducts matrix
from the previous example
⌈  ⌊ 
10.4 2.0 .6 
2.0 10.5 3.0 
.6 3.0 4.4 
⌉  ⌋ 
 The sums of squares are in the diagonal
(10.4, 10.5, 4.4), crossproducts
in the off diagonal.
 Now call the columns Y, X_{1},
and X_{2}
 Since we used deviation scores:
 Substitute S_{xx} for X'X
 Substitute S_{xy} for X'Y
 Therefore, B =
(S_{xx})^{1}S_{xy}
⌈ ⌊ 
10.5 3.0 
3.0 4.4 
⌉^{1}
⌋ 
⌈ ⌊ 
2.0 .6 
⌉ ⌋ 
= 
⌈ ⌊ 
0.19 0.01 
⌉ ⌋ 
 Those are the regression coefficients for
this example.
Lecture 9a: MULTIPLE REGRESSION ANALYSIS
IN R
 Simple regression has 1 predictor. Multiple
regression has 2 or more predictors.
 In this example, we want to predict
physical endurance based on
 Age.
 Years engaged in active exercise.
# Statistics One, Lecture 9a, example script
# Multiple regression analysis
# We are assuming normal distribution,
# linear relationships among variable,
# homoscedasticity, reliability and validity.
endur < read.table( "stats1.ex.04.txt",
header = T )
# Scatterplot
plot( endur$endurance ~ endur$age, main = "Scatterplot",
ylab = "Endurance", xlab = "Age" )
abline( lm( endur$endurance ~ endur$age ),
col="blue" )
plot( endur$endurance ~ endur$activeyears, main =
"Scatterplot", ylab = "Endurance",
xlab = "Active years" )
abline( lm( endur$endurance ~ endur$activeyears ),
col="red" )
# Regression analyses (unstandardized)
model1 = lm( endur$endurance ~ endur$age )
summary( model1 )
model2 = lm( endur$endurance ~ endur$activeyears )
summary( model2 )
model3 = lm( endur$endurance ~ endur$age
+ endur$activeyears )
summary( model3 )
Lecture 9b: MULTIPLE REGRESSION ANALYSIS,
STANDARDIZED
 New script, this time with standardized regression
coefficients. Then compare the two models.
 To get standardized regression coefficients,
use the "scale" function.
 To compare models, use the "anova" function
(analysis of variables).
# Statistics One, Lecture 9b, example script
# Multiple regression analysis "standardized"
# Same data as Lecture 9a.
# Regression analyses (standardized)
model1.z = lm( scale( endur$endurance )
~ scale( endur$age ) )
summary( model1.z )
model2.z = lm( scale( endur$endurance )
~ scale( endur$activeyears ) )
summary( model2.z )
model3.z = lm( scale( endur$endurance )
~ scale( endur$age )
+ scale( endur$activeyears ) )
summary( model3.z )
# Model comparisons
comp1 = anova( model1.z, model3.z )
comp1
comp2 = anova( model2.z, model3.z )
comp2
ASSIGNMENT THREE CODE
# Statistics One, Week Three Assignment
# Run correlation and multiple regression
on "daa.03.txt". Then answer 10 questions.
round to 2 significant digits.
assign03 < read.table( "daa.03.txt",
header = T)
# Question 1
# Correlation between age and
endurance
cor( assign03$age, assign03$endurance )
# .13
# Question 2
# Unstandardized regression coefficient
for age, predicting endurance
assign03.q2 = lm( assign03$endurance ~ assign03$age )
summary( assign03.q2 )
# .13
# Question 3
# Standardized regression coefficient
for age, predicting endurance
assign03.q3.z = lm( scale( assign03$endurance )
~ scale( assign03$age ) )
summary( assign03.q3.z )
# .13
# Question 4
# Unstandardized regression coefficient
for age, predicting endurance (while also using
activeyears to predict endurance)
assign03.q4 = lm( assign03$endurance
~ assign03$age + assign03$activeyears )
summary( assign03.q4 )
# .26
# Question 5
# Standardized regression coefficient
for age, predicting endurance (while also using
activeyears to predict endurance)
assign03.q5.z = lm( scale( assign03$endurance )
~ scale( assign03$age )
+ scale( assign03$activeyears ) )
summary( assign03.q5.z )
# .24
# Question 6
# Correlation between activeyears
and endurance
cor( assign03$activeyears, assign03$endurance )
# .34
# Question 7
# Unstandardized regression coefficient
for activeyears, predicting endurance
assign03.q7 = lm( assign03$endurance
~ assign03$activeyears )
summary( assign03.q7 )
# .76
# Question 8
# Standardized regression coefficient
for activeyears, predicting endurance
assign03.q8.z = lm( scale( assign03$endurance )
~ scale( assign03$activeyears ) )
summary( assign03.q8.z )
# .34
# Question 9
# Unstandardized regression coefficient
for activeyears, predicting endurance (while also
using age to predict endurance)
assign03.q9 = lm( assign03$endurance
~ assign03$activeyears + assign03$age )
summary( assign03.q9 )
# .92
# Question 10
# Standardized regression coefficient
for activeyears, predicting endurance (while also
using age to predict endurance)
assign03.q10.z = lm( scale( assign03$endurance )
~ scale( assign03$activeyears )
+ scale( assign03$age ) )
summary( assign03.q10.z )
# .40
Lecture 10a: MEDIATION ANALYSIS
REGRESSION METHOD
 Suppose we have a correlation between two
variable. We are looking for a third variable
or mechanism that links them. The intent is to
demonstrate causality, when randomized
experiments aren't possible.
 Moderation: a moderator is a third variable
that has control over a correlation or
relationship between two variables.
 An example:
 X: Psychological trait
 Y: Behavioral outcome
 M: Mechanism
 Diversity of life experience
 Z: Moderator (ZAP! or ZING!)
 SocioEconomicStatus (SES)
 We want to know why extraversion is
correlated with happiness?
 If X and Y are correlated then we can use
regression to predict Y from X
 If X and Y are correlated BECAUSE of the
mediator M then ( X ⇒ M ⇒ Y ):
 Y = B_{0} + B_{1}M
+ e
&
 M = B_{0} + B_{1}X
+ e
or
 Y = B_{0} + B_{1}M
+ B_{2}X + e
 What will happen to the predictive value
of X
 In other words, will B_{2}
be significant? If the mediator is
valid, B_{2} should be
insignificant.
 A mediator variable (M) accounts for some
(partial mediation) or all (full
mediation) of the relationship between
X and Y.
 CAUTION! Correlation does not imply causation!
There is a big difference between statistical
mediation and true causal mediation.
 Run three regression models
 lm( Y ~ X )
 Regression coefficient for X should
be significant.
 lm( M ~ X )
 Regression coefficient for X should
be significant.
 lm( Y ~ X + M )
 Regression coefficient for M should
be significant.
 If M is very significant, X would
have to be insignificant.
 Back to the example. Assume N = 188.
 Participants surveyed and asked to report:
 Happiness (happy).
 Extraversion (extra).
 Diversity of life experiences (diverse).
 Assume all are scored on a scale
from 1  5.
 Results of the first two models:
 happy = 2.19 + .28(extra).
 happy = 2.19 at the intercept, where
extraversion is zero. As extraversion
increases (per unit), add .28 units of
happy.
 diverse = 1.63 + .28(extra).
 For both, regression coefficient
for X (extra) is statistically
significant, p < .05.
 happy = 1.89 + .22(extra)
+ .19(diverse).
 ALL regression coefficients statistically
significant.
 This is partial mediation because the
direct effect on happy, (extra) only dropped
a little after the mediator (diverse) was
added into the regression equation. Happy did
not drop to zero.
Lecture 10b: MEDIATION ANALYSIS
PATH ANALYSIS METHOD
 Standard notation for Path include:
 Rectangles: Observed (manifested) variables
(X, Y, M) (predictor, outcome,
mediator).
 Circles: Unobserved variables (e).
 Triangles: Constants.
 Arrows: Associations.
 In the diagrams, variables Y and M are said
to be endogenous because the cause of variation
are inside the model.
 We are saying nothing about causes for our
predictor, X. It is said to be exogenous,
outside the model.
 Label the paths have been labeled a, b, c,
c'.
 The Sobel test
 z = (B_{a} * B_{b}) /
√(B_{a}²
* SE_{b}²)
+ (B_{b}²
* SE_{a}²)
 The null hypothesis
 The indirect effect is zero.
 (B_{a} * B_{b}) = 0
 Mediation analysis is more powerful with
 true independent variables
 the incorporation of time i.e.
one thing follows another).
 Otherwise, who is to say that happiness
doesn't cause extroversion.
Lecture 11a: MODERATION ANALYSIS
 Moderation: Introducing a new variable
that can completely change the picture,
or the relationship between two
variables.
 It might be the context of X and Y.
 The same example:
 X: Psychological trait
 Y: Behavioral outcome
 M: Mechanism
 Diversity of life experience
 Z: Moderator (ZAP! or ZING!)
 SocioEconomicStatus (SES)
 Perhaps, only rich extravert's are happier.
 A moderator variable (Z) has influence over
the relationship between X and Y.
 Suppose X and Y are positively
correlated.
 The moderator (Z) can change
that.
 If X and Y are correlated then we can use
regression to predict Y from X
 Y = B_{0} + B_{1}X
+ e
 B_{1} is the slope of X
to Y.
 If there is a moderator, Z, then
B_{1} will depend on Z.
 The relationship between X and Y
changes as a function of Z.
 A quick example:
 In two decades of working
testing, the professor has seen a
correlation between SAT's and working
memory capacity.
 Not at Princeton.
 They all have high SAT's, there is
nothing to correlate. There isn't
a representative sample.
 The type of University moderates the
relationship between WMC and SAT.
 Y = B_{0} + B_{1}X
+ B_{2}Z
+ B_{3}(X*Z) + e
 The B_{0} and B_{1}
are the standard formula. The
B_{2} and B_{3}
are the moderator.
 We can test for moderation with one
regression model
 lm( Y ~ X + Z + X*Z )
 Need to create a new column in
our data.frame for (X*Z).
 Let's call it PRODUCT.
 Back to the extrovert example. To simplify,
let's make SES categorical:
 SES = 1 = HIGH SES
 SES = 0 = LOW SES
 Results before adding PRODUCT (main or
first order affects).
 Ŷ = B_{0}(1) +
B_{1}(EXTRA) +
B_{2}(SES)
 Ŷ = 3.04 + .039(EXTRA) +
0.00(SES)
 Results after adding PRODUCT (main or
first order affects).
 Ŷ = B_{0}(1) +
B_{1}(EXTRA) +
B_{2}(SES) +
B_{3}(PRODUCT)
 Ŷ = 3.88 + .20(EXTRA) +
1.69(SES) + 0.47(PRODUCT)
 In this case, a low SES gives you a negative
correlation between extraversion and
happiness. High SES displays a positive
relationship.
 SES moderates the relationship between
extraversion and happiness.
Lecture 11b: CENTERING AND DUMMY CODING
 To center means to put our variable predictors (X)
in deviation form.
 X_{c} = X  M
 For every predictor variable, the mean
will now be zero.
 The regression constant (intercept)
is the predicted score on Y when all the
X's are zero.
 The regression constant now becomes
meaningful and easy to interpret.
 Why center?
 Conceptual
 Suppose
 Y = child's language development.
 X1 = mother's vocabulary.
 X2 = child's age.
 The intercept, B_{0}, is the
predicted score on Y when all X's
are zero.
 If X = zero is meaningless, or impossible,
B_{0} will be difficult to
interpret.
 If X = zero is the average then
B_{0} (intercept) is easy
to interpret.
 Now mother's vocabulary and child's
age are the average vocabulary
and age. The intercept is means
based and meaningful.
 But this is for simple regression only.
 There is another step for moderation analysis.
Remember how to interpret multiple regression
coefficients in multiple regression.
 The regression coefficient B_{1}
is the slope for X1 assuming an average
score on X2.
 No moderation implies the B_{1}
is consistent across the entire
distribution of X2.
 However, moderation implies the
B_{1} is NOT consistent across
the entire distribution of X2.
 Where in the distribution of X2 is
B_{1} most representative?
 The best place would be in the
middle of the distribution of
X2
 If it's centered, we're already
there.
 In this case, the regression constant
and the variable slopes will change,
but not the moderator slope.
 Also, the variable slopes are more
meaningful. They are the median
slopes, not the end slopes.
 Statistical
 The product X1*X2 can become highly
correlated with one of the predictors,
X1 or X2.
 It can result in
multicollinearity. That's a
bad thing.
 Summary
 If we center our predictors, we can
run a moderation analysis in a more
appropriate fashion.
 Now we can look at the moderation effects
by running sequential regression. There
are two ways to see if it is significant.
 Look at the main effects.
 Look at the moderation effect:
 Evaluate the regression
coefficient (slope)
for the product term. Is it
significant.
 ΔR² from Model 1 to
Model 2, which means do an Ftest
on the percentage of variance
explained on model one versus
model two.
 Dummy Coding: A system to code categorical
predictors in a regression analysis.
 What if SES in the earlier example had
more than two values (0 and 1).
 Example (faculty salary)
 IV: Area of research
 Cognitive
 Social
 Behavioral neuroscience
 Cognitive neuroscience
 DV: # of publications.
 Each Psychology professor is in
one of these four areas of
research, and how many
publications does each
researcher have in each
area.
 Question, do cognitive psychology
professors publish more than
social psychology professors?
 Is there a significant difference
in number of publications across
psychology departments?
 Add in a dummy variable into
the data.frame for each area
of research, 1 (4 areas get
3 columns). In this case
they are C1, C2, and C3
 Pick one group as
the reference group. We pick
cognitive. It has zeros for all
the codes.
 Each of the others gets a one
in one of the codes. Each is
a different code
(variable).
 The regression model is now:
 Ŷ = B_{0} +
B_{1}(C1) +
B_{2}(C2) +
B_{3}(C3)
 As there were a different number
of professors in each area, the
numbers aren't exact. You can
improve on this with "unweighted
effects" coding or "weighted
effects" coding.
Lecture 11c: MODERATION ANALYSIS EXAMPLE 2
 Faculty Salary Example
 DV = salary
 IVs
 # of publications
 Department
 Psychology
 Sociology
 History
 The question is, does department moderate
the relationship between publications and
between salary?
 Center continuous predictor.
 Dummy code categorical predictor.
 Create dummy codes for department.
 Regular dummy coding with Psych
as the reference group.
 Create moderation terms.
 Run sequential regression in 2 steps.
 Main effects.
 Moderation effect.
 Regression model: Before moderation.
 Ŷ = B_{0} +
B_{1}(PUBS.C) +
B_{2}(C1) +
B_{3}(C2)
 Regression model: Moderation.
 Ŷ = B_{0} +
B_{1}(PUBS.C) +
B_{2}(C1) +
B_{3}(C2) +
B_{4}(C1*PUBS.C) +
B_{5}(C2*PUBS.C)
 The data did show that the department you
are in, moderates between publications
and salary.
 To find out if the History or Sociology
slopes are significant, of if the
difference between them is, recode
to make a different reference group
rerun the analysis.
Lecture 12a: MEDIATION ANALYSIS IN R
 Write a script in R to test for mediation.
 Three regression analyses.
 Outcome = Predictor.
 Predictor = Mediator.
 Outcome = Predictor + Mediator.
 Three regression analyses.
 lm( Y ~ X )
 lm( Y ~ M )
 lm( Y ~ X + M )
 Fictional data
 Outcome (Y).
 Predictors (X, M).
 Extraversion (X).
 Diversity of life
experience (M).
# Statistics One, Lecture 12a, example script
# Mediation Analysis
# X is extraversion
# Y is happiness
# M is diversity of life experience
# Sobel test requires multilevel package.
install.packages( "multilevel" )
library( psych )
library( multilevel )
med < read.table( "stats1.ex.05.txt",
header = T )
# Test shape of distribution and univariate normal
assumptions
describe( med )
layout( matrix( c( 1,2,3 ), 1, 3, byrow = TRUE ) )
hist( med$happy )
hist( med$extra )
hist( med$diverse )
#Print scatter.plots to test linear and
homoscedasticity assumptions
layout( matrix( c( 1,2,3 ), 1, 3, byrow = TRUE ) )
plot( med$happy ~ med$extra )
abline( lm( med$happy ~ med$extra ) )
plot( med$diverse ~ med$extra )
abline( lm( med$diverse ~ med$extra ) )
plot( med$happy ~ med$diverse )
abline( lm( med$happy ~ med$diverse ) )
# Conduct three regression analyses
model.12a.1 = lm( med$happy ~ med$extra )
model.12a.2 = lm( med$diverse ~ med$extra )
model.12a.3 = lm( med$happy ~ med$extra +
med$diverse )
summary( model.12a.1 )
summary( model.12a.2 )
summary( model.12a.3 )
# Sobel test (is indirect path statistically
significant?)
indirect.12a = sobel( med$extra, med$diverse,
med$happy )
indirect.12a
Lecture 12b: MODERATION ANALYSIS IN R
 Fictional data
 Outcome (Y)
 Predictors (X, Z)
 Extraversion (X)
 SocioEconomic Status (SES) (Z)
# Statistics One, Lecture 12b, example script.
# Moderation analysis.
# X is extraversion.
# Y is happiness.
# Z is SES.
mod.12b < read.table( "stats1.ex.06.txt",
header = T )
# Data for mod.12b$mod variable is in not
# calculated here. It is in the data file.
no.mod.model.12b = lm( mod.12b$happy ~ mod.12b$extra
+ mod.12b$ses )
mod.model.12b = lm( mod.12b$happy ~ mod.12b$extra
+ mod.12b$ses + mod.12b$mod )
summary( no.mod.model.12b )
summary( mod.model.12b )
# Compare models.
anova( no.mod.model.12b, mod.model.12b )
Lecture 13a: STUDENT'S tTEST
 If we have randomized controlled experiments, we
don't need the regression models from the last
lectures. We can do ttests.
 Two means can be compared using a ttest.
 In this lecture we cover 4 variations of the
ttest.
 ztest
 ttest (single sample)
 ttest (dependent)
 ttest (independent)
 Overview
 z = (observed  expected) / SE.
 t = (observed  expected) / SE.
 SE: Standard error.
 When to use z and t?
 z
 When comparing a sample mean to a
population mean and the standard
deviation of the population is
known.
 Single sample t
 When comparing a sample mean to a
population mean and the standard
deviation of the population is
not known.
 Dependent samples t
 When evaluating the difference between
two related samples.
 Independent samples t
 When evaluating the difference between
two independent samples.
 observed 
expected  SE 
z  Sample mean 
Population mean  SE for a mean 
t (single sample) 
Sample mean 
Population mean  SE for a mean 
t (dependent) 
Difference  Difference 
SE for a difference 
t (independent) 
Difference  Difference 
SE for a difference 
 σ: population standard deviation.
 μ: population mean
 SD: sample standard deviation
 M: sample mean
 SE: standard error
 SE_{m}: standard error for a mean
 SE_{md}: standard error for
a difference (dependent)
 SE_{Difference}: standard error
for a difference (independent)
 Exact pvalue depends on:
 Directional or nondirectional test?
 df, degrees of freedom (different
tdistributions for different sample
sizes).
 Single sample t: Compare a sample mean to a
population mean
 t = ( M  μ ) / SE_{m}
 SE²_{m} = SD² / N
 SE_{m} = SD / SQRT( N )
 SD² = ∑( X  M )² /
( N  1 ) = SS / df = MS
 Example:
 Suppose it takes rats just 2 trials to learn
how to navigate a maze to receive a food
reward.
 A researcher surgically lesions part of the brain
and then test the rats in the maze. Is the
number of trials to learn the maze significantly
more than 2?
 Effect size (Cohen's d)
Lecture 13b: DEPENDENT & INDEPENDENT
tTESTS
 Single sample t compares a sample mean to a population
mean. Dependent t is a comparison of the same people
in two different conditions. Independent compares
two separate groups of people.
 Dependent means t
 The formulas are actually the same as the single
sample t but the raw scores are difference
scores, so the mean is the mean of the difference
scores and SE_{m} is based on the standard
deviation of the difference scores.
 Independent means t
 Compares two independent groups
 For example, males and females, control and
experimental, drug group and placebo group,
etc.
 t = (M_{1}  M_{2}) /
SE_{Difference}
 SE²_{Difference} =
SE²_{m1} +
SE²_{m2}
 SE²_{m1} =
SD²_{Pooled} / N1
 SE²_{m2} =
SD²_{Pooled} / N2
 SD²_{Pooled} =
(df_{1} /
df_{Total}) * (SD²_{1}) +
(df_{2} /
df_{Total}) * (SD²_{2})
 Notice that this is just a
weighted average of the sample variances.
Lecture 14a: GENERAL LINEAR MODEL
 ANOVA: analysis of variance.
 GLM is the mathematical framework used in many common
statistical analyses, including multiple regression
and ANOVA.
 ANOVA is typically presented as distinct from
multiple regression but IS a multiple
regression.
 It's a special case where the predictors are
orthogonal (not correlated).
 What they are is conditional or
categorical.
 The two main characteristics of GLM are
 Linear: pairs of variables are assumed to have
linear relations.
 Additive: if one set of variables predict another
variable, the effects are thought to be
additive.
 This doesn't mean we can't test nonlinear or
nonadditive effects (interaction).
 GLM can accommodate such tests, for example
 Transformation of variables
 Transform so nonlinear becomes
linear.
 Moderation analysis
 Fake the GLM into testing nonadditive
effects (product scores used in
moderation analysis).
 Simple regression
 Y = B_{0} + B_{1}X_{1}
+ e
 Y = faculty salary
 X1 = years since PhD
 Multiple regression
 Y = B_{0} + B_{1}X_{1}
+ B_{2}X_{2}
+ B_{3}X_{3}
+ e
 Y = faculty salary
 X1 = years since PhD
 X2 = # of publications
 X3 = (years x pubs)
 In the last example, "years since PhD" and "#
of publications" are continuous variables. But
gender or race are conditional or
categorical variables.
 Oneway ANOVA
 Y = B_{0} + B_{1}X_{1}
+ e
 Y = faculty salary
 X1 = gender
 Factorial ANOVA
 Y = B_{0} + B_{1}X_{1}
+ B_{2}X_{2}
+ B_{3}X_{3}
+ e
 Y = faculty salary
 X1 = gender
 X2 = race
 X3 = interaction (gender x race)
 ANOVA: Appropriate when the predictors (independent
variable or IVs) are all categorical and the outcome
(DV) is continuous.
 Most common application is to analyze data
from randomized controlled experiments.
 More specifically, randomized experiments that
generate more than 2 means
 If only 2 means then use:
 Independent ttest.
 Dependent ttest.
 If more than 2 means then use:
 If they are independent groups, then between
groups ANOVA.
 If the same subjects measured repeatedly,
then repeated measures ANOVA.
 This is analogous to the independent
ttest and the dependent ttest.
 NHST may accompany ANOVA
 The test statistic is the Ftest (ratio)
rather than a t statistic.
 F = systematic variance / unsystematic
variance
 Like ttest, the Ftest has a family of Fdistributions
 The distribution to assume depends on
 Number of subjects per group.
 Number of groups.
 This is a ratio of variances, it
can't be less than zero.
 The expected value under the null hypothesis
is one.
 This is analogous to the Z distribution
of the normal curve.
 If we get a number like 5 we are way out
on the tail of the curve, we will reject
the null hypothesis and claim a significant
effect.
Lecture 14b: ONEWAY ANOVA
 Example: working memory training, how many sessions
they trained, outcome is the gain in their scores.
 Fratio
 F = systematic variance / unsystematic variance
 F = betweengroups variance /
withingroups variance
 F = MS_{Between} / MS_{Within}
 F = MS_{a} / MS_{s/a}
(s/a: subject within group).
 MS_{a} = SS_{a} /
df_{a}
 MS_{s/a} = SS_{s/a} /
df_{s/a}
 SS_{a} = n ∑(Y_{j} 
Y_{T})²
 Y_{j} are the treatment means.
 Y_{T} is the grand mean.
 SS_{s/a} = n ∑(Y_{ij} 
Y_{j})²
 Y_{ij} are individual scores.
 Y_{j} are the treatment means.
 Degrees of freedom
 df_{A} = a  1.
 df_{s/a} = a(n  1).
 df_{Total} = N  1.
 Effect size
 η² (etasquared) is analogous
to R² in multiple regression, percentage
of variance explained in the outcome
variable.
 η² = SS_{A} /
SS_{Total}.
 Assumptions:
 DV is continuous.
 DV is normally distributed.
 Homogeneity of variance.
 Withingroups variance is equivalent for
all groups.
 Posthoc test. If levene.test() is significant
then homogeneity of variance assumption has been
violated.
 Conduct comparisons using a restricted error
term.
Lecture 14c: FACTORIAL ANOVA
 Two IVs (independent variables) or treatments.
 One continuous DV or dependent variable
(response).
 IV = driving difficulty (easy, difficult).
 IV = conversation difficulty (none, easy,
difficult).
 DV = errors made in driving simulator.
 Three hypotheses can be tested in this one
experiment:
 More errors in the difficult simulator?
 More errors with more difficult
conversations?
 More errors due to the interaction of these
factors?
 We will calculate three Fratios.
 Terms
 Main effect: the effect of one IV averaged
across the levels of the other IV.
 Interaction effect: the effect of one IV
depends on the other IV (the simple effects
of one IV change across the levels of the
other IV).
 Simple effect: the effect of one IV at a
particular level of the other IV,
if significant.
 Main effects and interaction effect are independent
from one another (orthogonal).
 Note that this is different from studies that
don't employ an experimental design.
 For example, in MR, when predicting
faculty salary, the effects of
publications and years since the Ph.D.
were correlated..
 Remember GLM (general linear model).
 Multiple regression looked like
 Y = B_{0} + B_{1}X_{1}
+ B_{2}X_{2}
+ B_{3}X_{3} + e
 Y = faculty salary.
 X1 = years since Ph.D.
 X2 = # of publications.
 X3 = (years x pubs).
 Factorial ANOVA
 Y = B_{0} + B_{1}X_{1}
+ B_{2}X_{2}
+ B_{3}X_{3} + e
 Y = faculty salary.
 X1 = gender.
 X2 = race.
 X3 = interaction (gender x race).
 We want 3 Fratios.
 F_{A} = MS_{A} /
MS_{S/AB}
 F_{B} = MS_{B} /
MS_{S/AB}
 F_{AxB} = MS_{AxB} /
MS_{S/AB}
 MS (mean squared)
 MS_{A} = SS_{A} /
df_{A}
 MS_{B} = SS_{B} /
df_{B}
 MS_{AxB} = SS_{AxB} /
df_{AxB}
 MS_{S/AB} = SS_{S/AB} /
df_{S/AB}
 df (degrees of freedom)
 df_{A} = a  1
 a = levels of driving difficulty
(21).
 df_{B} = b  1
 b = levels of conversation difficulty
(31).
 df_{AxB} = (a  1)(b  1)
 df_{S/AB} = ab(n  1)
 number of group x subjects in a group
minus one.
 df_{Total} = abn  1 = N  1
 Follow up tests.
 Main effects
 Posthoc tests
 Only where we have more than 2 groups.
 Significant interaction effects
 It's not enough to show there are significant
interaction effects, we have to show where
they are coming from.
 Analysis of simple effects
 Conduct a series of oneway ANOVAs
 How, where, when and why are the
simple effects different.
 For example, we could conduct 3
oneway ANOVAs comparing high
and low spans at each level of
other IV.
 Effect size
 Complete η²
 η² = SS_{effect} /
SS_{total}
 Partial η²
 η² = SS_{effect} /
(SS_{effect} + SS_{S/AB})
 The assumptions underlying the factorial ANOVA are the
same as for the oneway ANOVA.
 DV is continuous.
 DV is normally distributed.
 Homogeneity of variance.
 Back to the driving example.
 Strayer and Johnson (2001) conducted an
experiment to examine the effect of talking
on a cellphone on driving.
 They tested subjects in a driving simulator.
 Here's the interesting part...
 They manipulated the difficulty of the
driving,
 AND the difficulty of the conversation.
 To manipulate driving difficulty, they simply made the
driving course in the simulator more or less
difficult.
 To manipulate conversation difficulty, they included
two "talking" conditions:
 In one, the subject simply had to repeat what
they heard on the other line of the phone.
 In the other, the subject had to think of and
then say a word beginning with the last letter
of the last word spoken on the phone.
 e.g. if you hear "ship" then respond a word
that begins with p  "pear".
 And there was a notalking condition
(control).
 Again, a summary of the design.
 IV 1 = driving difficulty (easy, difficult).
 IV 2 = conversation difficulty (none, easy,
difficult).
 DV = errors made in driving simulator.
 We would refer to this as a 2 x 3 factorial
design. We will analysis it with a 2 x 3
between groups ANOVA.
 Followup tests
 Posthoc tests
 Need to conduct posthoc tests on the
conversation IV.
 No need for driving difficulty because
there's only 2 levels.
 Simple effects
 Simple effect of conversation at each
level of driving difficulty.
OR
 Simple effect of driving difficulty at
each level of conversation.
 Simple effects analysis to explain the interaction
variable.
 Look at the effect of one independent variable
at each level of the other. Either effect
of driving difficulty for each conversation
condition, or vice versus.
 F_{a at bk} =
MS_{a at bk} /
MS_{S/AB}
 MS_{a at bk} =
SS_{a at bk} /
df_{a at bk}
 df_{a at bk} = a  1
 SS_{a at bk} =
n ∑(Y_{jk} 
Y_{Bk})²
 We choose simple effect of driving difficulty at
each level of conversation.
 No conversation.
 F = MS_{A at b1} /
MS_{S/AB}
 F = 324.9 / 6.267 = 51.84
 Easy.
 F = MS_{A at b2} /
MS_{S/AB}
 F = 592.9 / 6.267 = 94.61
 Difficult.
 F = MS_{A at b3} /
MS_{S/AB}
 F = 1000 / 6.267 = 159.57
 Effect sizes.
 No conversation.
 η² = SS_{A at b1} /
(SS_{A at b1} +
SS_{S/AB})
 η² = 324.9 / (324.9 + 150.4)
= .68
 Easy.
 η² = SS_{A at b2} /
(SS_{A at b2} +
SS_{S/AB})
 η² = 529.9 / (529.9 + 150.4)
= .80
 Difficult.
 η² = SS_{A at b3} /
(SS_{A at b3} +
SS_{S/AB})
 η² = 1000 / (1000 + 150.4)
= .87
Lecture 15a: STUDENT TTEST AND ANOVA IN R.
 Is it possible for adults to enhance their
intelligence by training their working
memory.
 Population
 Sample
 Random selection from the population.
 Representative of the population.
 Procedure
 Treatment group engaged in memory training
for a half hour every day for weeks.
 IQ change
 All subjects completed a test of
intelligence before and after
training.
 Results were reported two different ways.
 First approach
 IV: WM training (training vs. control)
 DV: Score on intelligence test (pre and
post).
 Second approach
 IV: WM training (number of sessions).
 DV: Gain on intelligence test
(post  pre).
 Write a script in R to illustrate dependent t
 IV: Pre and post
 dependent ttest, within a group, same
subject two different conditions.
 DV: Score
 Write a script in R to illustrate independent t
 IV: Training group (control, training)
 DV: Gain
 Write a script in R to illustrate ANOVA
 IV: Training group (8, 12, 17, 19)
 DV: Gain
Lecture 15b: TTEST IN R.
 Write a script in R
 dependent t
 IV: Pre and post.
 DV: Score.
 independent t
 IV: Training group (control, training).
 DV: Gain.
# Statistics One, Lecture 15b,
example script.
# Write ttests in R using working memory
training data
# Dependent t, IV: Pre & post.
# DV: Score.
# Independent t. IV: Training group (control, training).
# DV: Gain.
# Read data into R
lec15b = read.table( "stats1.ex.07.txt",
header = T )
# Print descriptive statistics for the variables
# in wm by training condition (cond)
describeBy( lec15b, lec15b$cond )
# Create subsets of data for control and training
conditions
lec15b.c = subset( lec15b, lec15b$train == "0" )
lec15b.t = subset( lec15b, lec15b$train ==
"1" )
# We need internal descriptive statistics to calculate
effect size.
lec15b.c.out = describe( lec15b.c )
lec15b.t.out = describe( lec15b.t )
lec15b.c.out
lec15b.t.out
# Dependent ttests
t.test( lec15b.c$pre, lec15b.c$post,
paired = T )
t.test( lec15b.t$pre, lec15b.t$post,
paired = T )
# Cohen's d for dependent ttest
d.c = (lec15b.c.out[ 4, 3 ]) /
(lec15b.c.out[ 4, 4 ])
d.t = (lec15b.t.out[ 4, 3 ]) /
(lec15b.t.out[ 4, 4 ])
d.c
d.t
# Independent ttests
t.test( lec15b$gain ~ lec15b$train, var.equal = T )
# Cohen's d for independent ttests
lec15b.pooled.sd = (79/118 * lec15b.t.out[ 4, 4 ])
+ (39/118 * lec15b.c.out[ 4, 4 ])
d.ct = (lec15b.t.out[ 4, 3 ] 
lec15b.c.out[ 4, 3 ]) / lec15b.pooled.sd
d.ct
Lecture 15c: ANOVA IN R.
 This time we will use the working memory training,
using "number of sessions" and gain on intelligence
tests.
 Because we have 4 groups, we want to use a oneway
between groups ANOVA.
# Statistics One, Lecture 15c, example
script.
# Write ANOVA in R, using memory training data.
# IV: WM training (number of sessions).
# DV: Gain on intelligence test (post  pre).
# Oneway between groups ANOVA.
library( psych )
library( gclus)
library( multilevel )
library( lawstat)
# aov( DV ~ IV )
lec15c.aov.model = aov( lec15b.t$gain ~ lec15b.t$cond )
summary( lec15c.aov.model )
lec15c.aov.table = summary( lec15c.aov.model )
# Effect size for ANOVA
lec15c.ss = lec15c.aov.table[[1]]$"Sum Sq"
lec15c.eta.sq = lec15c.ss[1] / ( lec15c.ss[1] +
lec15c.ss[2] )
lec15c.eta.sq
# Posthoc tests
TukeyHSD( lec15c.aov.model )
library( lawstat )
# levene.test( DV, IV, mean )
levene.test( lec15b.t$gain, lec15b.t$cond,
location=c("mean") )
Lecture 16a: BENEFITS OF REPEATED MEASURES
(within subjects) ANOVA
 Less cost (fewer subjects required). Every subject
is exposed to every condition in your experiment.
 More subjects is more statistical power.
 Variance across subjects may be systematic.
 If so, it will not contribute to the error
term.
 Error term
 F_{A} = MS_{A} /
MS_{AxS}
 MS and F
 MS_{A} = SS_{A}
/ df_{A}
 MS_{AxS} = SS_{AxS}
/ df_{AxS}
 F = MS_{A} / MS_{AxS}
 Example
 A classic memory and learning paradigm is
AB/AC paired associate learning.
 Subjects learn a list of paired associates,
AB.
 Then they learn another list, AC.
 Subsequently, recall is impaired when cued
with A and asked to recall B.
 This is known as retroactive interference.
 df
 df_{A} = a  1 = 3  1 = 2
 df_{s} = n  1 = 6  1 = 5
 df_{AxS} = (a  1)(n  1) = 10
 df_{Total} = (a)(n)  1 = 17
 MS and F
 MS_{A} = 700 / 2 = 350
 MS_{AxS} = 2.67 / 10 = .267
 F = 1310.86
 Posthoc tests
 The error term MS_{AxS} is NOT
appropriate.
 Need to calculate a new error term
based on the conditions that are
being compared.
 F_{Ψ} = MS_{ΨA} /
MS_{ΨAxS}
 MS_{ΨA} = SS_{ΨA}
/ df_{ΨA}
 MS_{ΨAxS} = SS_{ΨAxS}
/ df_{ΨAxS}
 Where Ψ is a subset of.
 If you do lots of those corrections, one for every
pair, you can use the Bonferroni method.
 Sphericity assumption
 Homogeneity of variance.
 Homogeneity of correlation.
 Homogeneity of correlation can be tested
with Mauchly's test.
 If significant then report the pvalue from
one of the corrected tests
 GreenhouseGeisser
 HuynFeldt
Lecture 16b: RISKS OF REPEATED MEASURES ANOVA
 Risks
 Order effects.
 Practice effects.
 Fatigue effects.
 Counterbalancing scheme to deal with these
effects.
 Missing data.
 Counterbalancing
 Consider a simple design with just two
condition, A1 and A2.
 One approach is a Blocked Design
 Subjects are randomly assigned to one
of two "order" conditions.
 Another approach is a Randomized Design.
 Conditions are presented randomly in
a mixed fashion.
 A1, A2.
 A2, A1, A1, A2, A2, A1, A2, ...
 Now suppose a = 3 and a blocked design.
 There are 6 possible orders (3!).
 A1, A2, A3.
 A1, A3, A2.
 A2, A1, A3.
 A2, A3, A1.
 A3, A1, A2.
 A3, A2, A1.
 To completely counterbalance, subjects would be
randomly assigned to one of 6 order conditions.
 The number of conditions needed to completely
counterbalance becomes large with more
conditions..
 With many levels of the IV a better approach is
to use a "Latin Squares" design.
 Latin Squares designs aren't completely
counterbalanced but every condition appears
every position at least once.
 For example, if a = 3, then
 A1, A2, A3.
 A2, A3, A1.
 A3, A1, A2.
 This isn't perfect but it is very close. The
number of orderings you need is the number of
levels of independent variables.
 Missing data.
 Relative amount of missing data.
 Pattern of missing data.
 How much missing data is a lot?
 There are no hard and fast rules.
 A rule of thumb is
 Less than 10% on any one variable is OK.
 Greater than 10% is not OK.
 Is the pattern random or lawful?
 For any variable of interest (X) create a new
variable (XM).
 XM = 0 if X is missing.
 XM = 1 if X is not missing.
 Conduct a ttest with XM as the IV and X as
the DV.
 If significant then pattern of missing data
may be lawful.
 Missing data ~ Remedies
 Drop all cases without a perfect profile.
 Drastic.
 Use only if you can afford it.
 Keep all cases and estimate the values of
the missing data points.
 There are several options for how to
estimate values.
 Missing data ~ Estimation methods
 Insert the mean.
 Conservative.
 Decreases variance and covariance. That
means you don't have much to work with.
 Regressionbased estimation.
 More precise than using the mean but
 Confusion often arises over which variables
to use as predictors in the regression
equation.
 You must clearly spell out what you did and
how you did it. Other wise it smells of
manufactured data to fit your goals.
Lecture 17a: MIXED FACTORIAL ANOVA
 Mixed factorial designs.
 One IV (independent variable) is manipulated
between groups.
 One IV is manipulated within groups.
 Repeated measures variable.
 What's new?
 Partitioning SS.
 Formula for F_{A}, F_{B},
F_{AxB}.
 Error term for posthoc tests.
 Approach to simple effects analyses.
 Assumptions.
 df
 df_{A} = a  1
 df_{B} = b  1
 df_{AxB} = (a  1)(b  1)
 df_{S/A} = a(n  1)
 df_{BxS/A} = a(b  1)(n  1)
 df_{Total} = (a)(b)(n)  1
 MS
 MS_{A} = SS_{A} /
df_{A}
 MS_{B} = SS_{B} /
df_{B}
 MS_{AxB} = SS_{AxB} /
df_{AxB}
 MS_{S/A} = SS_{S/A} /
df_{S/A}
 MS_{BxS/A} = SS_{BxS/A} /
df_{BxS/A}
 F
 F_{A} = MS_{A} /
MS_{S/A}
 F_{B} = MS_{B} /
MS_{BxS/A}
 F_{AxB} = MS_{AxB} /
MS_{BxS/A}
 Posthoc tests on main effects.
 Posthoc tests on the betweengroups IV
are performed in the same way as with a
oneway ANOVA.
 Posthoc tests on repeated measures IV
are performed in the same way as with a
oneway repeated measures ANOVA.
 Pairwise comparisons with Bonferroni
correction.
 Simple effects analyses.
 Must choose one approach or the other.
To report both is redundant.
 (a) Simple effects of the between groups
IV.
 (b) Simple effects of the repeated
IV.
 Simple effects of the between groups IV.
 Simple effect of A at each level of B.
 F_{A.at.b1} =
MS_{A.at.b1} /
MS_{S/A.at.b1}
 Simple comparisons use the same error term
(assuming homogeneity of variance).
 Simple effect on the repeated measures IV.
 Simple effect of B at each level of A.
 F_{B.at.a1} =
MS_{B.at.a1} /
MS_{BxS/A.at.a1}
 Assumptions.
 Each subject provides b scores.
 Therefore,there are
 b variances.
 ( b * (b + 1) / 2 )  b
covariances (correlations).
 e.g., if b = 3 then 3 covariances.
 e.g., if b = 4 then 6 covariances.
 Betweengroups assumptions.
 The variances do not depend upon the
group (homogeneity of
variance).
 If levene.test() is violated,
calculate a new restricted
error term.
 The covariances do not depend upon the
group (homogeneity of covariance).
 Box's test of equality of
covariance matrices.
 If violated then report
GreenhouseGeisser or
HuynhFeldt values. Alternatively,
consider a moderation analysis.
 Withinsubjects assumptions.
 Sphericity: the variances of the different
treatment scores (b) are the same and the
correlations among pairs of treatment
means are the same.
 If violated, report GreenhouseGeisser
or HuynhFeldt values.
Lecture 17b: MIXED FACTORIAL ANOVA IN R
 The experimental design of another working memory example.
 Mixed factorial design, 3x2, (Ax(BxS)).
 Between groups IV (independent variable).
 Memory task (words, sentences, stories).
 Within groups IV.
 Phonological similarity (similar, dissimilar).
 DV (dependent variable).
 Percentage of words recalled.
 Hypotheses.
 Main effect of memory task?
 Memory task (words, sentences, stories).
 Main effect of similarity?
 Phonological similarity (similar, dissimilar).
 DV (dependent variable).
 Three Fratios
code for "eta_squared.R"
eta.2 = function(aov.mdl, ret.labels = FALSE)
{ eta.2vector = c()
labels = c()
for ( table in summary(aov.mdl) )
{ # each block of factors
SS.vector = table[[1]]$"Sum Sq"
# table is a list with 1 entry,
# but you have to use [[1]] anyway
last = length(SS.vector)
labels = c(labels, row.names(table[[1]])[last])
# all but last (error term)
for (SS in SS.vector[last])
{ #all but last entry (error term)
new.etaval = SS / (SS + SS.vector[last])
eta.2vector = c(eta.2vector, new.etaval)
}
}
if (ret.labels) return(data.frame(eta.2 =
eta.2vector, row.names = labels))
return(eta.2vector)
}
 Using Linux, I needed to make these corrections to code for
stats1.ex.08.R
 eta_squared.R: on the course web sight, see the post
Discussion Forums, Problems running the script for Lect 17b
sample Script in R
 Error: "task not found"
 He puts the column of the data.frame but he
doesn't always list which data.frame.
 [task=="W"] needs to be [e1sr$task=="W"]
 lines 22, 23, 28 and 29
 e1sr, that is not the letter "L", it's the
number one.
 Same data.frame issue with the describeBys
 recall[task=="W"] should be e1sr$recall[e1sr$task=="W"]
 stim[task=="W"] should be e1sr$stim[e1sr$task=="W"]
 lines 39  41
 my version requires describeBy(), not describe.by()
 This is linux. The levene.test() in package(car) only
returns the median. To get the mean, use
package(lawstat)
 In the lecture, Professor Conway said they we have to reverse
S and D in "levels=c("S","D")", line 12. Otherwise, the graph
will be backwards. I did, it didn't change anything. The bars
are still backwards from the ones he shows on his slides.
#########
# stats1.ex.08.R, Macnamara, Moore, & Conway (2011),
# Experiment 1, Serial recall
#########
library(psych)
library(lawstat)
source(file="eta_squared.R")
e1sr < read.table("stats1.ex.08.txt", header = T)
# Omnibus analysis is a 3x2 mixed factorial with task
# and stimuli as the independent variables and serial
# recall as the dependent variable. The three levels
# of task are word span, reading span, and story span.
# The two levels of stimuli are phonologically similar
# and phonologically dissimilar.
stim = factor(e1sr$stim,levels=c("S","D"))
#reverse levels (for graphs like the article)
aov.e1sr = aov(e1sr$recall ~ (e1sr$task*e1sr$stim) +
Error(factor(e1sr$subject)/e1sr$stim))
summary(aov.e1sr)
eta.2(aov.e1sr, ret.labels=TRUE)
# levene.test()
levene.test( e1sr$recall, e1sr$task, location=c("mean") )
# Simple effects analysis for simple span (i.e., word span)
aov.e1srw = aov(e1sr$recall[e1sr$task=="W"] ~
e1sr$stim[e1sr$task=="W"] +
Error(factor(e1sr$subject[e1sr$task=="W"]) /
e1sr$stim[e1sr$task=="W"]))
summary(aov.e1srw)
eta.2(aov.e1srw, ret.labels=TRUE)
# Simple effects analysis for complex
# span (this is a 2x2 mixed factorial)
aov.e1srnw = aov(e1sr$recall[e1sr$task!="W"] ~
e1sr$task[e1sr$task!="W"] *
e1sr$stim[e1sr$task!="W"] +
Error(factor(e1sr$subject[e1sr$task!="W"]) /
e1sr$stim[e1sr$task!="W"]))
summary(aov.e1srnw)
eta.2(aov.e1srnw, ret.labels=TRUE)
#######
#Graph
# Bar plot
wspan = describeBy(e1sr$recall[e1sr$task=="W"],
group = e1sr$stim[e1sr$task =="W"], mat = T)
rspan = describeBy(e1sr$recall[e1sr$task=="R"],
group = e1sr$stim[e1sr$task =="R"], mat = T)
sspan = describeBy(e1sr$recall[e1sr$task=="S"],
group = e1sr$stim[e1sr$task =="S"], mat = T)
graphme = cbind(Words = wspan$mean, Sentences =
rspan$mean, Stories = sspan$mean)
rownames(graphme) = c("Phonologically Similar",
"Phonologically Dissimilar")
se = cbind(wspan$se, rspan$se, sspan$se)
bp = barplot( graphme, beside = TRUE,
ylim = c(0,1), space = c(0, .5), legend.text = TRUE,
args.legend = c(x = "topright") )
abline(h=0)
for (ii in 1:3)
{ arrows(bp[1, ii], graphme[1,ii]  se[1, ii],
y1 = graphme[1,ii] + se[1, ii], angle = 90, code = 3)
arrows(bp[2, ii], graphme[2,ii]  se[2, ii],
y1 = graphme[2,ii] + se[2, ii], angle = 90, code = 3)
}
Lecture 18: SUMMARY
 Experimental design.
 Randomized controlled experiments.
 Different types of designs in ANOVA lectures.
 Descriptive statistics.
 Histograms.
 Summary statistics.
 Correlation & regression.
 If two variables are correlated, I can use
one to predict the other.
 Measurement.
 Bias.
 Reliability.
 Validity.
 Multiple regression.
 Matrix algebra.
 Estimation of coefficients.
 Mediation.
 Moderation.
 Central limit theorem.
 Sampling.
 Sampling error.
 Standard error.
 NHST.
 Flaws.
 Alternatives:
 Effect size.
 Model comparison.
 Tests to compare means.
 Use ttests if you only have two means.
 Use ANOVA if there are more than two.
WEEK 6 ASSIGNMENT
 We now return to cognitive training. Suppose we
conducted a training experiment in which subjects
were randomly assigned to one of three conditions:
 Working Memory training (WM)
 Physical Exercise training (PE)
 Designed Sports training (DS)
 Further assume that we measured spatial reasoning ability
before and after training, using two separate measures:
 SR1 (pretraining)
 SR2 (posttraining)
 Fictional data are available in the file:
DAA.05.txt
or daa.05.txt on this computer.
 Write an R script to answer the following questions (the
main analysis should be a 3x2 mixed factorial ANOVA). All
values should be given with two decimal places.
 Our theory predicts a significant interaction between time
and condition. What is the F ratio for the interaction?
 One assumption of the analysis is homogeneity of variance.
What is the F ratio for Levene’s test?
 What is the F ratio for the simple effect of time for
the WM group?
 What is the F ratio for the simple effect of time for
the PE group?
 What is the F ratio for the simple effect of time for
the DS group?
# Statistics One, Assignment 6
# A 3x2 mixed factorial ANOVA
# to analysis 3 working memory conditions: WM, PE, DS,
# in pre (SR1) and post (SR2) training.
library( psych )
library( lawstat)
source( file="eta_squared.R" )
ass6 < read.table( "daa.05.txt", header = T )
aov.ass6 = aov( ass6$SR ~ ( ass6$condition * ass6$time )
+ Error( factor( ass6$subject ) / ass6$time) )
summary( aov.ass6 )
levene.test( ass6$SR, ass6$condition, location=c("mean") )
aov.ass6.WM = aov( ass6$SR[ ass6$condition=="WM" ] ~
ass6$time[ ass6$condition=="WM" ] +
Error( factor( ass6$subject[ ass6$condition=="WM" ] )
/ ass6$time[ ass6$condition=="WM" ] ) )
aov.ass6.PE = aov( ass6$SR[ ass6$condition=="PE" ] ~
ass6$time[ ass6$condition=="PE" ] +
Error( factor( ass6$subject[ ass6$condition=="PE" ] )
/ ass6$time[ ass6$condition=="PE" ] ) )
aov.ass6.DS = aov( ass6$SR[ ass6$condition=="DS" ] ~
ass6$time[ ass6$condition=="DS" ] +
Error( factor( ass6$subject[ ass6$condition=="DS" ] )
/ ass6$time[ ass6$condition=="DS" ] ) )
summary( aov.ass6.WM )
summary( aov.ass6.PE )
summary( aov.ass6.DS )
FINAL EXAM
Grade: 16 out of 20 
Question 1 
Sally scored a 70 on her math
quiz but the mean for the class was 40. If the
standard deviation was 6 then what was Sally’s
zscore? 
 5 
X  5 
 6 
 6 
Question 2 
According to the central limit
theorem, the shape of the distribution of sample
means is almost always: 
 negatively skewed 
 uniform 
X  normal 
 positively skewed 
Question 3 
The mean of the distribution of
sample means equals: 
X  the mean of the population 
 1 
 the mean of the sample 
 0 
Question 4 
The standard deviation of the
distribution of sample means is called: 
 likely error 
X  standard error 
 percent error 
 special error 
Question 5 
When plotting correlational data,
the appropriate graph to use is a: 
 line graph 
X  scatterplot 
 bar plot 
 histogram 
Question 6 
Which of the following rvalues
indicates the strongest relationship between two
variables? 
 .10 
 65 
X  .89 
 .10 
Question 7 
What is the slope in the following
regression equation? Y = 2.69X – 3.92 
 2.69 
 3.92 
X  2.69 
 3.92 
Question 8 
When we square the correlation
coefficient to produce r2, the result is equal to
the: 
 proportion of variance in Y not
accounted for by X 
 sum of squared residuals 
 standard error 
X  proportion of variance in Y accounted
for by X 
Question 9 
What value is expected for the t
statistic if the null hypothesis is true? 
 1 
 2 
X  0 
 1.96 
Question 10 
What happens to the tdistribution as
the sample size increases? 
 The distribution becomes uniform 
 The distribution is unaffected 
X  The distributions appears more and more
like a normal distribution 
 The distribution appears less and less like
a normal distribution 
Question 11 
Degrees of freedom (df) for the single
sample ttest is equal to: 
 N + 1 
 N 
X  N  1 
 the square root of N 
Question 12 wrong 
In an independent ttest, what is
the standard error of the difference? 
 the pooled standard deviation 
 the standard deviation of the
distribution of sample mean differences 
 the standard deviation of the
distribution of sample means 
X  the standard deviation of the
sample means 
Question 13 wrong 
How many subjects were included in an
independent samples ttest if a researcher reports
t(20) = 3.68 
 20 
X  21 
 18 
 22 
Question 14 
In a standard regression analysis,
if the unstandardized regression coefficient is 2
and the standard error of the regression coefficient
is 4 then what is the corresponding
tvalue? 
 .5 
Question 15 
What type of graph can you examine
to test the assumption that a bivariate relationship
is linear? 
 scatterplot 
Question 16 wrong 
What are the total degrees of
freedom in a 3 x 3 factorial ANOVA with 10 subjects
per group? 
 9 
Question 17 
How many regression analyses need
to be conducted to test for mediation? 
 3 
Question 18 
Should you use a ztest or a ttest
when the population mean and the population standard
deviation are both known? 
 ztest 
Question 19 
Besides homogeneity of variance,
what does sphericity assume? 
 homogeneity of correlation 
Question 20 wrong 
What procedure can you use to
correct for multiple comparisons in repeated
measures ANOVA? 
 counterbalancing 
