Fall Exam
• What is in the exam:
– Chapters 1-9
– Lecture Slides 01-08 (up to and including Linear Regression)
– Assignments 1-2
PSYC 6130, PROF. J. ELDER
1
Fall Term Material
Descriptive Statistics
•
•
Basics
– Scales of measurement
– Means, medians, modes
– Continuous vs. discrete variables
– Standard deviations, SQI, range
– Independent and dependent variables
– Skewness, kurtosis
– Experimental vs. correlational research
•
– Populations vs. samples
– Z scores
Tables and Graphs
– SEM
– f, cf, rf, crf, cpf
– Central Limit Theorem
– Grouped frequency distributions
– Apparent vs real limits
– Percentiles, percentile ranks, linear
interpolation
PSYC 6130, PROF. J. ELDER
The Normal Distribution
– Probability rules
– Summation notation
•
Central Tendency and Variability
3
Inferential Statistics: Hypothesis Tests of the Mean
– Introduction to NHT
– Type I and Type II errors
– One-tailed vs two-tailed tests
– 6-step procedure
– One-sample z test
– One-sample t test
– t Test for two independent sample means
– Homogeneity of variance
– The Matched t Test
– Going beyond null hypothesis testing
– Planning experiments (power)
– Effect size
– Confidence intervals
PSYC 6130, PROF. J. ELDER
4
Correlation and Regression
– Pearson’s r
– Significance and power for Pearson’s r
– Fisher Transform
– Regression formulae
– Explained and unexplained variance
– Coefficients of determination, non-determination
– Homoscedasticity
– Confidence intervals for predictions
– Partialing out
PSYC 6130, PROF. J. ELDER
5
Some Important Topics
Basic Procedure for Statistical Inference
1. State the hypothesis
2. Select the statistical test and significance level
3. Select the sample and collect the data
4. Find the region of rejection
5. Calculate the test statistic
6. Make the statistical decision
PSYC 6130, PROF. J. ELDER
7
NHT for Two Small Independent Samples
By analogy with one-sample NHT, we might approximate
the standard errors  X1 and  X2 by the sample standard errors s X1 and s X2 .
Unfortunately, the resulting sampling distribution of the difference of the means
is not straightforward to analyze.
So what do we do?
If we can assume homogeneity of variance (the two populations have the same variance),
then there is a statistic that follows the t distribution and is simple to analyze.
PSYC 6130, PROF. J. ELDER
8
Pooled Variance
Pooled variance is s
2
p
n1  1 s12   n2  1 s22


 n1  1   n2  1
df1s12  df2s22

df1  df2
Note that the pooled variance is a weighted sum of the sample variances.
The weights are proportional to the size of each sample
(Bigger samples are more reliable estimators of the common variance)
df2
df1
sp2
s12
PSYC 6130, PROF. J. ELDER
9
s22
Summary: t-Tests for 2 Independent Sample Means
n1, n2  100 n1
s2 Test
n2 s1
statistic
s X2 1  X2
df



t
s12 s22

n1 n2
WelchSatterthwaite



t
sp2
n1  n2  2
n1

sp2
n2



t
s12 s22

n1 n2
n1  n2  2



t
s12 s22

n1 n2
n1  n2  2



z
s12 s22

n1 n2
NA



z
s12 s22

n1 n2
NA



z
s12 s22

n1 n2
NA



z
s12 s22

n1 n2
NA
PSYC 6130, PROF. J. ELDER
10
Separate Variances t Test
• If
– Population variances are different (suggested by substantially
different sample variances)
AND
– Samples are small
AND
– Sample sizes are substantially different
• Then
– Pooled variance t statistic will not be correct.
• In this case, use separate variances t test
PSYC 6130, PROF. J. ELDER
11
Separate Variances t Test
X

t
1

 X 2   1  2 
s X1  X2
where sX2 1 X2  sX2 1  sX2 2
•
This t statistic is well approximated by a t distribution.
•
Unfortunately, calculating the appropriate df is difficult.
•
SPSS will calculate the Welch-Satterthwaite approximation for df as
part of a 2-sample t test:
s

df  
2
X1
s X4 1
df1
s

2
X2

2
s X4 2
df2
PSYC 6130, PROF. J. ELDER
12
More on Homogeneity of Variance
• How do we decide if two sample variances are different enough to
suggest different population variances?
• Need NHT for homogeneity of variance.
– F-test
• Straightforward
• Sensitive to deviations from normality
– Levene’s test
• More robust to deviations from normality
• Computed by SPSS
PSYC 6130, PROF. J. ELDER
13
Levene’s Test: Basic Idea
1. Replace each score X1i , X2i with its absolute deviation from the sample mean:
d1i | X1i  X1 |
d 2i | X 2i  X 2 |
2. Now run an independent samples t-test on d1i and d2i :
t
d1  d 2
sd1 d2
SPSS reports an F-statistic for Levene’s test
• Allows the homogeneity of variance for two or more variables to be tested.
• We will cover the F distribution in the winter term.
PSYC 6130, PROF. J. ELDER
14
Independent or Matched?
• Application of the Independent-Groups t test depended
on independence both within and between groups.
• There are many cases where it is wise, convenient or
necessary to use a matched design, in which there is a
1:1 correspondence between scores in the two samples.
• In this case, you cannot assume independence between
samples!
• Examples:
– Repeated-subject designs (same subjects in both samples).
– Matched-pairs designs (attempt to match possibly important
attributes of subjects in two samples)
PSYC 6130, PROF. J. ELDER
15
Better alternative:
The matched t-test using the direct difference method
A3
A4 A4-A3
72 80
70 80
69 93
83 88
88 93
88 93
87 88
88 93
85 100
85 100
70 80
72 90
60 80
83 75
81 83
68 75
36 93
80 100
65 83
65 83
41 75
73 88
68 75
Mean
SD
n
8
10
24
5
5
5
1
5
15
15
10
18
20
-8
2
7
57
20
18
18
34
15
7
73
86
13
14
23
8
23
13
23
PSYC 6130, PROF. J. ELDER
t
X  0
s/ n
t 
16
D  0
sD / n
Limitations of NHT
• Criticisms of NHT date from the 1930s.
– Null hypothesis is rarely true.
– The real question is not about the existence of an effect, but
about the nature of the effect:
• What is the direction of the effect?
• What is the size of the effect?
• How important is it?
• What are the underlying mechanisms (theory)?
PSYC 6130, PROF. J. ELDER
17
Magnitude of Effect
•
A problem with a point estimate is that it suggests a certainty we do not really have.
•
A more complete and useful description of the magnitude of the effect is provided by
a confidence interval.
e.g., s X1  X2
2.632 2.852


 0.044"
7586 7777
 The 95% confidence interval for 1 -2




  X1  X 2  z /2s X1  X2 , X1  X 2  z /2s X1  X2 
X  69.87 "
s  2.63 "
n  7586
 0.86" 1.96  0.044",0.86" 1.96  0.044"
 [0.77",0.95"]
X  69.01"
s  2.85 "
n  7777
PSYC 6130, PROF. J. ELDER
18
Importance of Effect
• However it is often more meaningful to compare the treatment
effect to the overall variation in the measured variable:
d
X  69.87 "
s  2.63 "
n  7586
1  2

X1  X 2
sp
df1s12  df2s22
where s 
df1  df2
2
p
7585  2.632  7776  2.85 2

7585  7776
 7.53
X  69.01"
s  2.85 "
n  7777
Thus sp  2.74"
d
PSYC 6130, PROF. J. ELDER
19
0.86 "
 0.31
2.74 "
Example Effect Sizes
0.4
Group 1
0.4
Group 2
0.2
0
-5
0.4
0.2
0
d=.5
0.4
0.2
0
-5
PSYC 6130, PROF. J. ELDER
0
-5
5
0
d=1
5
0
d=4
5
0.2
0
d=2
0
-5
5
20
Planning a Study
• There are many considerations that go into planning an
experiment or study.
• Here we focus on the statistical considerations.
• Some possible questions:
– How many samples (e.g., subjects) will I need for my study?
– I already know that I will only have access to n samples
(subjects). Will this be enough?
• Answering these questions depends on understanding
the relationship between sample size, effect size, and
statistical power.
PSYC 6130, PROF. J. ELDER
21
Standardized Distributions of the Difference of the Means
NHD
AHD
0.4
Probability p(t)
0.3

Power 1  
0.2
1

0.1
0
-4
-2
0
E[t | H0 ]
2
tcrit
t
PSYC 6130, PROF. J. ELDER
22
4

n
d
2
6
8
Estimating Power
If we have an estimate of the expected t value  ,
we can estimate the power 1  
 Pr(t  tcrit   | E[t ]  0)
(Non-central t distribution)
(Central t distribution)
Pr(t)
Pr(t)
1   Pr(t  tcrit | E[t ]   )

tcrit
1 

1 

tcrit  
0
Expected t value
PSYC 6130, PROF. J. ELDER
23
Calculating Power from Sample Size and Effect Size

-
tcrit
Sample size n
+
  E [t ]
Effect Size d
+

PSYC 6130, PROF. J. ELDER
n
d
2
24
+
Power 1  
Planning Experiments
• Planning experiments may involve estimating any one of these
variables given knowledge or assumptions about the other two:
n, d  power:  
n
d
2
You have already decided on the size of your sample, and you have an estimate of the effect size.
What is the power of your experiment?
 
d, power  n : n  2  
d 
2
You have an estimate of the effect size, and know the minimum power you want.
What sample size do you need?
n, power  d : d 
2

n
You have already decided on the size of your sample, and you know the power you want.
What does the size of the effect have to be to give you this power?
PSYC 6130, PROF. J. ELDER
25
Correlation
Standard Definition of Correlation (Sample)
Recall that the sample variance sX2 of X is

s E X X

2
X

2

 1
XX
 n  1 

2
We define the sample covariance sXY of X and Y as





1
s XY  E  X  X Y  Y  
X  X Y Y

n 1

The Pearson correlation r between X and Y is then given by
r
s XY
s X sY
PSYC 6130, PROF. J. ELDER
27
Computational Formula
covariance
For a population:
For a sample:
z z

x
N
y
( X  

x
)(Y  y )
N x y

 XY   
N
1
 zx zy   ( X  X )(Y  Y )  N  1
r 
N 1
(N  1)sx sy
unbiased covariance
PSYC 6130, PROF. J. ELDER
28
x
y
 x y
  XY  NXY 
s x sy