1. Psych 706: stats II
Class #4

2. AGENDA In-Class Exam next Tuesday
Homework Assignment #2 due next Tuesday
Review z-score calculations and ANCOVA
Correlation
Any Exam-Related Questions you have
SPSS tutorial: Correlation

3. Z-score SKEWNESS/KURTOSIS CALCULATIONS
In addition to just eyeballing histograms to look for normality, you can determine whether the skewness and kurtosis in your data are within the realm of what you’d expect within the normal distribution
Within Normal Limits!
Non-normal
Non-normal
z = -1.96
p < .05 two-tailed
cutoff
z = +1.96
p < .05 two-tailed
cutoff

4. EXAMPLE Exam Performance z-scores Skewness -.373/.238 = -1.56
Kurtosis -.852/.472 = -1.81
Within Normal Limits!
Non-normal
Non-normal
Conclusion
Both z-scores within normal distribution so we’re good!
NORMALITY ASSUMPTION HOLDS
z = -1.96
p < .05 two-tailed
cutoff
z = +1.96
p < .05 two-tailed
cutoff

5. EXAMPLE Exam Anxiety z-scores Skewness -2.012 /.238 = -8.45
Kurtosis 5.192/.472 = 11
Within Normal Limits!
Non-normal
Non-normal
Conclusion
Both z-scores way outside the normal distribution
NORMALITY ASSUMPTION VIOLATED!
z = -1.96
p < .05 two-tailed
cutoff
z = +1.96
p < .05 two-tailed
cutoff

6. ANCOVA Review 2+ Groups and want to use covariate in your analysis…
to make error in model smaller
Thus making your group effect larger in size/more significant
Covariate…
shares variance with your dependent variable similarly across groups (homogeneity of regression slopes)
cannot differ between groups (independence of covariate and group)

7. Homogeneity of regression slopes
Relationship between covariate (Partner’s Libido) and dependent variable (Libido) need to be in the same direction for all three groups
In this case, they are not so this assumption is violated and Partner’s Libido cannot be used as a legit covariate

8. Independence of covariate and group
Groups cannot differ on the covariate
You test this by making the covariate the dependent variable in a one-way ANOVA with group as the independent variable
If ANOVA p < .05, the groups differ on the covariate
In this case, the ANOVA is not significant so this assumption is met.
However we already know based on the previous slide that Partner’s Libido cannot be used as a legit covariate

9. FOR ANCOVA, just like ANOVA/T tests:
Check for Normality for each group separately
Check for Homogeneity of Variance between groups

10. ANY questions about material we’ve Covered thus far?

11. Correlation (and Regression)
Outcome = Predictor Variable(s) + Error
Y = bX E
Population parameter: How closely variables are associated with outcome
Cannot assume causation
Assumes linear relationship between predictor and outcome variables
This is the simplest model – we’ll talk about more complicated versions soon

12. variance
The average amount that individual data points for variable X vary from their mean

13. Covariance
The average amount that two variables X and Y deviate from their means in a similar direction
What is the main difference between the the variance vs. covariance formulas?

14. Covariance Positive value = both variables deviate similarly
Negative value = one variable goes up while the other goes down
Problem: This measure depends on units of measurement of variables and is difficult to interpret b/c it’s not standardized
Covariance

15. WAIT, How do you standardize AGAIN?
Score minus the mean, divided by the standard deviation

16. Pearson’s product-moment correlation coefficient
Solution: Divide by standard deviation of both variables (sort-of like the z-formula that you use to standardize and compare scores with different units)!

17. Pearson’s product-moment correlation coefficient
Varies between -1 and +1
Two types
Bivariate = how much 2 variables covary
Partial = how much 2 variables covary while “controlling” for 1+ other variables
Measure of effect size:
+/- .1 = small
+/- .3 = medium
+/- .5 = large

18. Pearson’s product-moment correlation coefficient
Testing significance of a correlation:
Is r different from zero (a.k.a. no relationship between the two variables)?
Use t-test calculation

19. Pearson’s r Assumptions: Linearity and Normality
Normality only needed if you plan to compute significance tests or use non- bootstrapped confidence intervals
Graph histograms to look for normality
Graph scatterplots to look for outliers/linearity
Outcome variable must be INTERVAL scale
If assumptions violated, you can either use Spearman’s rho or Kendall’s tau

20. Exam anxiety Pre-test exam anxiety (0-100) Exam performance (%)
Do these variables share significant variance?

21. Pearson’s bivariate r: SPSS OUTPUT
“A significant negative relationship exists between anxiety and performance, r(103)= -.441, p<.001, such that as anxiety increases, performance decreases.”

22. SPEARMAN’S RHO Non-parametric correlation
Rank-order the data and then calculate Pearson’s formula on those ranks
Examples:
Winning a spelling bee (1st, 2nd, 3rd, 4th)
Grades (A+/-, B+/-, C+/-, D+/-, F)
Acceptance order for graduate school in Clinical Psych Ph.D. for Queens College

23. SPEARMAN’S RHO: SPSS OUTPUT
“A significant negative relationship exists between anxiety and performance, ρ(103)= -.405, p<.001, such that as anxiety increases, performance decreases.”

24. Kendall’s tau Non-parametric correlation
Better estimate of population correlation than Spearman’s rho
Use instead of Spearman’s correlation when you have:
A small sample size
A bunch of tied ranks (many people with A-’s, several sergeants)

25. Kendall’s tau: SPSS OUTPUT
“A significant negative relationship exists between anxiety and performance, τ(103)= -.285, p<.001, such that as anxiety increases, performance decreases.”

26. TYPES of correlations (THREE variables: X1, X2, Y)
X1 and X2 = Independent variables
Y = Dependent variable
Zero-Order Bivariate = variance shared between X1 and Y, regardless of X2
Partial = unique variance shared between X1 and Y with X2 shared variance removed entirely
Part / Semi-Partial = unique variance shared between X1 and Y compared to all possible variance (including X2)

27. Correlation example
Major Depressive Disorders and Anxiety Disorders are highly comorbid
Research indicates that there are two specific dimensions of anxiety that might differ in behavior, cognition, and physiology
Anxious Apprehension (Worry)
Anxious Arousal (Panic)
Let’s do some correlations to see how much variance each dimension shares with depression symptoms!

28. Y X1 e a d c X2 Depression Worry Panic
Y = Depression, our Dependent Variable
We want to figure out all of the variance associated with the Y pie

29. Zero-order correlation: Depression and worry
Panic a c d Y X1 X2 e
How much variance depression and worry share
Regardless of panic symptoms
r between X1 and Y:
(a + d) / (a + d + c + e)

30. Zero-order correlation: Depression and panic
Worry
Panic a c d Y X1 X2 e
How much variance depression and panic share
Depression
Worry
Regardless of worry symptoms
r between X2 and Y:
(c + d) / (a + c + d + e)
Panic

31. partial correlation: depression and worry
Panic a c d Y X1 X2 e
How much UNIQUE variance depression and worry share,
with panic removed from the Depression pie entirely
Depression
Worry
Partial r
between X1 and Y:
a / (a + e)
Panic

32. partial correlation: Depression and panic
How much UNIQUE variance depression and panic share, with worry removed from the Depression pie entirely
Depression
Worry
Panic a c d Y X1 X2 e
Depression
Worry
Partial r
between X2 and Y:
c / (c + e)
Panic

33. Part/SEMI-PARTIAL correlation: depression and worry
Panic a c d Y X1 X2 e
How much UNIQUE variance depression and worry share with respect to
the ENTIRE Depression pie
Depression
Part r
between X1 and Y:
a /( a + c + d + e)
Panic

34. Part/SEMI-PARTIAL correlation: Depression and panic
Worry
Panic a c d Y X1 X2 e
How much UNIQUE variance depression and panic share with respect to the ENTIRE Depression pie
Depression
Part r
between X2 and Y:
c / (a + c + d + e)
Panic

35. Why COMPUTE a partial correlation?
Say you have three variables that are all significantly correlated with each other (using Bivariate correlation like Pearson’s r)
You might want to see if the relationship between 2 of the variables is due entirely or partially to both sharing important variance with the 3rd variable
Example: ↑ methamphetamine use is significantly correlated with ↑ antisocial behaviors
However, is it the shared variance with poverty that is driving these two variables to be correlated?
Rationale: ↑ poverty = ↑ drug use, and ↑ poverty = ↑ crime
Run partial correlation btw meth and antisocial, controlling for poverty!

36. Relationship between METH and ANTISOCIAL BEHAVIOR
Zero-order:
(a + b) / (a + b + c + d)
How much total variance do meth and antisocial share?
(Not including poverty)
r = .33, p=.04
METH
X2: POVERTY
ANTISOCIAL Y X1 a b c d
Partial:
a / (a + d)
Removing all shared
variance with poverty,
do meth and antisocial
uniquely share any leftover (residual) variance?
r = .21, p=.19
Poverty is driving the
correlation between
meth and antisocial behavior!

37. Why Compute a Part/ SeMI-PARTIAL CORRELATION?
It tells you the proportion of total variance that two variables uniquely share
Say you hypothesize that poverty in particular will share more variance with antisocial behavior than meth use
You could then compute part correlations to show whether your hypothesis was supported:
Poverty + antisocial unique variance / total variance
VERSUS
Meth + antisocial unique variance / total variance

38. Relationship between METH and ANTISOCIAL BEHAVIOR
X2: POVERTY
ANTISOCIAL Y X1 a b c d
Part/Semi-Partial:
a / (a + b + c + d)
What proportion of total variance is uniquely shared by meth and antisocial
r = .09, p=.47
Part/Semi-Partial:
c / (a + b + c + d)
What proportion of total variance is uniquely shared by poverty and antisocial
r = .55, p=.006
Poverty contributes more unique variance to the model than meth!

39. Are two correlations significantly different from each other?
Independent samples
Is the amount of shared variance between depression and worry SIMILAR…
For women and men?
For depressed and control groups?
Fisher’s r-to-z transformation
Plug in #s here:
Dependent samples
Within the depressed group, is the amount of shared variance SIMILAR…
Between depression and worry?
Between depression and panic?

40. Calculating effect size
Good news! You don’t need to calculate this stuff because it’s in your SPSS output!
Pearson’s r = standardized covariation between two variables
R-squared (R²) = r² proportion of variance shared by two variables, ranging from 0 to 1

41. Questions?