Simple Covariation
Focus is still on
‘Understanding the Variability”
With Group Difference approaches, issue
has been:
Can group membership (based on ‘levels of the IV’)
account for variability of the DV?
Information used was differences in ‘typical’ outcomes
across the levels of the IV.
Simple Covariation
Did typical outcomes differ enough to suggest
the presence of systematic variability?
Was variability in IV associated with variability in DV to a degree unlikely
to be due to ‘unsystematic’ variations?
How much of the variability has been
‘explained’, and how much has not (residual)
Simple Covariation
Now the focus is on the degree to which pairs of
variables from a common source covary.
(are systematic changes in one variable
associated with systematic changes in the other)
No longer looking at typical performance for the
group, now variability of both variables is at the
individual level.
If the two variables covary systematically, then
knowing one variable might ‘explain’ or ‘account for’
variability of the other
Simple Covariation
Source of paired scores can be any type of
entity: people, days, families, countries….
No longer categorize variables as IV and DV,
just two variables from the same ‘source’
Seek to measure the strength of the
relationship (covariation) between two
variables.
Simple Covariation
Correlation Coefficient is an index of the
relationship
All of these provide an index of ‘strength’ of the
relationship on a 0 – 1 ordinal scale
Some also provide ‘direction’ information, when
appropriate (+/-)
Simple Covariation
Correlation Coefficient is the index of the
relationship
Various forms, depending upon data
Pearson’s r – two interval/ratio variables
eta – one nominal, one interval/ratio variable
phi or Cramer’s V – two categorical variables
Spearman’s rho – two ordinal or one ordinal
and one interval variable (scores converted to ranks)
Not all provide meaningful direction information – but SPSS will still give sign
Simple Covariation
Common applications
Preliminary evidence, prior to controlled
experiment - If cause and Effect exists, covariation should
Assess degree of association/similarity among
variables – Is Cheerfulness the same as Agreeableness
Is Optimism related to Risk Taking
Develop prediction strategy – can SAT predict CollegeSuccess
Simple Covariation
Pearson’s Product Moment Coefficient (r)
Index of strength and direction of a
linear relationship
if two variables covary in a linear relationship, then
an individual’s relative position
(deviations from means)
on each variable should be similar
Simple Covariation
Pearson’s Product Moment Coefficient (r)
r = covariance/‘variance’ – refresh on calculation of variance
show connection to covariance
r = sum (zx * zy)/df (n-1)
r2 = shared variance (ratio scale)
coefficient of determination
Ho: r = 0, tested using a t-test with n-2 df
n = # of pairs of measures
Simple Covariation
Examine the relationship using scatter plot
Perceived Stress in the Past, and Expected Stress in the Future
No stress 0 to 56 Highest stress
Simple Covariation
Assumptions for Pearson’s r
interval/ratio data
independent observations (pairs)
each variable normally distributed (or not obviously not normal)
linear relationship (no evidence of clear nonlinear pattern)
bivariate normal distribution – (3 dimensional normal pile)
homoscedasticity (similar variability of Y at values of X)
Simple Covariation
Limiting conditions for Pearson’s r
bivariate outliers – reduces r if truly outlier on both variables
truncated range – effect depends upon actual relationship (linear or nonlinear)
Simple Covariation
With all data
Correlations
Perceived Stress
Last Month
Perceived Stress
Next Month
Pearson Correlation
Sig . (2-tailed)
N
Pearson Correlation
Sig . (2-tailed)
N
Perceived
Stress
Last Month
1
Perceived
Stress
Next Month
.738**
.000
176
176
.738**
1
.000
176
176
**. Correlation is sig nificant at the 0.01 level (2-tailed).
With two pairs removed
Limiting conditions for Pearson’s r
Correlations
bivariate outliers
truncated range
Perceived Stress
Last Month
Perceived Stress
Next Month
If try to ‘fit’ a straight line through the scatter-plot. How would
the 2 outliers impact the line?
Pearson Correlation
Sig . (2-tailed)
N
Pearson Correlation
Sig . (2-tailed)
N
Perceived
Stress
Last Month
1
Perceived
Stress
Next Month
.768**
.000
174
174
.768**
1
.000
174
174
**. Correlation is sig nificant at the 0.01 level (2-tailed).
Simple Covariation
Typical sequence in evaluating r
check assumptions
calculate r When reporting r, df are number of ‘pairs’ minus 2
assess statistical significance t-test for r=0
compute r2 Coefficient of determination
interpret strength and direction
discuss “effect size” – shared variance
Simple Covariation
If you wanted to interpret all of the r’s, you
would have 15 tests on the same individuals –
so Type 1 will be inflated. However, you may
only care about r’s for GREs with GPA Total,
so only 4 r’s are relevant. As always, balance
Type 1 and Type 2.
Listwise – must have score on every variable
Correlationsa
Undergrad GPA Total
Undergrad GPA Jr Sr
Years
GRE Verbal
GRE Quantitative
GRE Analytic
GRE Advanced Psych
Pearson Correlation
Sig. (2-tailed)
Pearson Correlation
Sig. (2-tailed)
Pearson Correlation
Sig. (2-tailed)
Pearson Correlation
Sig. (2-tailed)
Pearson Correlation
Sig. (2-tailed)
Pearson Correlation
Sig. (2-tailed)
Undergrad
Undergrad
GPA Jr Sr
GPA Total
Years
1.000
.746**
.
.000
.746**
1.000
.000
.129*
.044
.149*
.021
.229**
.000
.282**
.000
.
.176**
.006
.078
.229
.151*
.018
.263**
.000
GRE
GRE
Verbal
Quantitative
.129*
.149*
.044
.021
.176**
.078
GRE
Analytic
.229**
.000
.151*
.006
.229
.018
.000
1.000
.
.255**
.000
.276**
.000
.525**
.000
.255**
.000
1.000
.
.527**
.000
.444**
.000
.276**
.000
.527**
.000
1.000
.
.369**
.000
.525**
.000
.444**
.000
.369**
.000
1.000
.
**. Correlation is significant at the 0.01 level (2-tailed).
*. Correlation is significant at the 0.05 level (2-tailed).
a. Listwise N=242
GRE
Advanced
Psych
.282**
.000
.263**
Note sample size here, and on next page, from SAME data set!
Simple Covariation
Pairwise – included whenever have both scores for a coefficient
N’s much
lower in
column for
GRE Analytic –
why?
Correlations
Undergrad
GPA Total
Undergrad GPA Total
Undergrad GPA Jr Sr
Years
GRE Verbal
GRE Quantitative
GRE Analytic
GRE Advanced Psych
Pearson Correlation
Sig. (2-tailed)
N
Pearson Correlation
Sig. (2-tailed)
N
Pearson Correlation
Sig. (2-tailed)
N
Pearson Correlation
Sig. (2-tailed)
N
Pearson Correlation
Sig. (2-tailed)
N
Pearson Correlation
Sig. (2-tailed)
N
**. Correlation is significant at the 0.01 level (2-tailed).
1
.
393
.730**
.000
371
.160**
.001
393
.136**
.007
393
.280**
.000
296
.307**
.000
348
Undergrad
GPA Jr Sr
Years
GRE Verbal
.730**
.160**
.000
.001
371
393
1
.171**
.
.001
372
372
.171**
1
.001
.
372
399
.026
.225**
.615
.000
372
399
.156**
.259**
.009
.000
279
302
.251**
.502**
.000
.000
327
351
GRE
Quantitative
GRE Analytic
.136**
.280**
.007
.000
393
296
.026
.156**
.615
.009
372
279
.225**
.259**
.000
.000
399
302
1
.563**
.
.000
399
302
.563**
1
.000
.
302
302
.412**
.377**
.000
.000
351
263
N’s range from 263 to 399 using Pairwise
GRE
Advanced
Psych
.307**
.000
348
.251**
.000
327
.502**
.000
351
.412**
.000
351
.377**
.000
263
1
.
351
Simple Covariation
Pearson r vs. Spearman rho
Correlations
Ease of Return to Work
Colleagues' Acceptance
Customers' Acceptance
Pearson Correlation
Sig. (2-tailed)
N
Pearson Correlation
Sig. (2-tailed)
N
Pearson Correlation
Sig. (2-tailed)
N
Ease of
Return to
Colleagues'
Customers'
Work
Acceptance
Acceptance
1.000
.561**
.445**
.
.000
.000
1216
1216
1216
.561**
1.000
.608**
.000
.
.000
1216
1216
1216
.445**
.608**
1.000
.000
.000
.
1216
1216
1216
Difference based on whether you
were willing to consider rating scale:
Definitely no (1) to (9) Definitely yes
to be interval or ordinal
**. Correlation is significant at the 0.01 level (2-tailed).
Correlations
Spearman's rho
Ease of Return to Work
Colleagues' Acceptance
Customers' Acceptance
Correlation Coefficient
Sig. (2-tailed)
N
Correlation Coefficient
Sig. (2-tailed)
N
Correlation Coefficient
Sig. (2-tailed)
N
**. Correlation is significant at the .01 level (2-tailed).
Ease of
Return to
Colleagues'
Customers'
Work
Acceptance
Acceptance
1.000
.565**
.445**
.
.000
.000
1216
1216
1216
.565**
1.000
.601**
.000
.
.000
1216
1216
1216
.445**
.601**
1.000
.000
.000
.
1216
1216
1216
Simple Covariation
Covariation and causality
Conditions needed to infer Cause-Effect
1 two variables covary (covariation)
2 cause precedes the effect
3 other potential causes controlled
Simple Covariation
Covariation and causality
Conditions needed to infer Cause-Effect
1 two variables covary (covariation)
Correlation coefficients can provide a reasonable test of condition #1
Is there evidence for significant (systematic) covariation?
2 cause precedes the effect
3 other potential causes controlled
Simple Covariation
Covariation and causality
Conditions needed to infer Cause-Effect
1 two variables covary (covariation)
2 cause precedes the effect
Correlation does not directly deal with this condition – creating the…
Directionality problem
X Y or Y X - which of these is more likely to be true
Cross-lagged strategy – provides evidence to help decide
Simple Covariation
Covariation and causality
Cross-lagged strategy
Time 1
Var X (TV violence)
Var Y (Aggressive
Behaviors)
Simple Covariation
Covariation and causality
Cross-lagged strategy
Time 1
Var X (TV violence)
Time 2
Var X (TV violence)
Var Y (Aggressive
Behaviors)
Var Y (Aggressive
Behaviors)
Simple Covariation
Covariation and causality
Cross-lagged strategy
Time 1
Var X (TV violence)
Time 2
Var X (TV violence)
Y as Cause
X as Cause
Var Y (Aggressive
Behaviors)
Var Y (Aggressive
Behaviors)
Which direction of cause – effect receives stronger support
Simple Covariation
Covariation and causality
Conditions needed to infer Cause-Effect
1 two variables covary (covariation)
2 cause precedes the effect
3 other potential causes controlled
Because you simply select for or measure your variables, have less potential to isolate the
variables of interest from other extraneous variables – creating…
“Third” Variable Problem
The Solution – Partial Correlation
Simple Covariation
Covariation and causality
Partial correlation (pr)
Examine correlation of X & Y after ‘removing’ variation in
each that can be explained by variable Z
X
Y
Third variable problem exists when both
X and Y are related to Z, the Third
variable, so the covariation of X and Y is
the result of Z influencing both X and Y
Z
(correlation of the residuals for X and Y after
removing relationship with Z) – clearer after regression
Simple Covariation
Correlations
Investment
Model
Investments
Rating
Investment Model
Investments Rating
Investment Model
Commitment Rating
How long in months?
Pearson Correlation
Sig. (2-tailed)
N
Pearson Correlation
Sig. (2-tailed)
N
Pearson Correlation
Sig. (2-tailed)
N
1
.
71
.699**
.000
71
.341**
.004
71
Investment
Model
Commitment
Rating
.699**
.000
71
1
.
71
.349**
.003
71
How long
in months?
.341**
.004
71
.349**
.003
71
1
.
71
Women in dating relationships where there had been
physical abuse, were asked for rated Commitment to
her partner, Time in relationship, and Perceived
Investments in relationship
Commitment and How long
in months you have been
in the relationship are
correlated at +.349
**. Correlation is significant at the 0.01 level (2-tailed).
Correlations
Control Variables
Investment Model
Investments Rating
Investment Model
Commitment Rating
How long in months?
Correlation
Significance (2-tailed)
df
Correlation
Significance (2-tailed)
df
Investment
Model
Commitment
Rating
1.000
.
0
.165
.171
68
How long
in months?
.165
.171
68
1.000
.
0
When control for Investments made to
relationship, correlation reduced to +.165