Simple Covariation
description
Transcript of Simple Covariation
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 variableassociated 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 variableseta – one nominal, one interval/ratio variablephi or Cramer’s V – two categorical variablesSpearman’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 alinear 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 dfn = # 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
Limiting conditions for Pearson’s r
bivariate outliers
truncated range
Correlations
1 .738**
.000
176 176
.738** 1
.000
176 176
Pearson Correlation
Sig. (2-tailed)
N
Pearson Correlation
Sig. (2-tailed)
N
Perceived StressLast Month
Perceived StressNext Month
PerceivedStress
Last Month
PerceivedStress
Next Month
Correlation is significant at the 0.01 level (2-tailed).**.
Correlations
1 .768**
.000
174 174
.768** 1
.000
174 174
Pearson Correlation
Sig. (2-tailed)
N
Pearson Correlation
Sig. (2-tailed)
N
Perceived StressLast Month
Perceived StressNext Month
PerceivedStress
Last Month
PerceivedStress
Next Month
Correlation is significant at the 0.01 level (2-tailed).**.
With all data
With two pairs removed
If try to ‘fit’ a straight line through the scatter-plot. How would the 2 outliers impact the line?
Simple Covariation
Typical sequence in evaluating r
check assumptionscalculate 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
Correlationsa
1.000 .746** .129* .149* .229** .282**
. .000 .044 .021 .000 .000
.746** 1.000 .176** .078 .151* .263**
.000 . .006 .229 .018 .000
.129* .176** 1.000 .255** .276** .525**
.044 .006 . .000 .000 .000
.149* .078 .255** 1.000 .527** .444**
.021 .229 .000 . .000 .000
.229** .151* .276** .527** 1.000 .369**
.000 .018 .000 .000 . .000
.282** .263** .525** .444** .369** 1.000
.000 .000 .000 .000 .000 .
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 GPA Total
Undergrad GPA Jr SrYears
GRE Verbal
GRE Quantitative
GRE Analytic
GRE Advanced Psych
UndergradGPA Total
UndergradGPA Jr Sr
YearsGRE
VerbalGRE
QuantitativeGRE
Analytic
GREAdvanced
Psych
Correlation is significant at the 0.01 level (2-tailed).**.
Correlation is significant at the 0.05 level (2-tailed).*.
Listwise N=242a.
Listwise – must have score on every variable
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.
Note sample size here, and on next page, from SAME data set!
Simple Covariation
Correlations
1 .730** .160** .136** .280** .307**
. .000 .001 .007 .000 .000
393 371 393 393 296 348
.730** 1 .171** .026 .156** .251**
.000 . .001 .615 .009 .000
371 372 372 372 279 327
.160** .171** 1 .225** .259** .502**
.001 .001 . .000 .000 .000
393 372 399 399 302 351
.136** .026 .225** 1 .563** .412**
.007 .615 .000 . .000 .000
393 372 399 399 302 351
.280** .156** .259** .563** 1 .377**
.000 .009 .000 .000 . .000
296 279 302 302 302 263
.307** .251** .502** .412** .377** 1
.000 .000 .000 .000 .000 .
348 327 351 351 263 351
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
Undergrad GPA Total
Undergrad GPA Jr SrYears
GRE Verbal
GRE Quantitative
GRE Analytic
GRE Advanced Psych
UndergradGPA Total
UndergradGPA Jr Sr
Years GRE VerbalGRE
Quantitative GRE Analytic
GREAdvanced
Psych
Correlation is significant at the 0.01 level (2-tailed).**.
Pairwise – included whenever have both scores for a coefficient
N’s range from 263 to 399 using Pairwise
N’s much lower in column for GRE Analytic – why?
Simple Covariation
Correlations
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
Pearson Correlation
Sig. (2-tailed)
N
Pearson Correlation
Sig. (2-tailed)
N
Pearson Correlation
Sig. (2-tailed)
N
Ease of Return to Work
Colleagues' Acceptance
Customers' Acceptance
Ease ofReturn to
WorkColleagues'Acceptance
Customers'Acceptance
Correlation is significant at the 0.01 level (2-tailed).**.
Correlations
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
Correlation Coefficient
Sig. (2-tailed)
N
Correlation Coefficient
Sig. (2-tailed)
N
Correlation Coefficient
Sig. (2-tailed)
N
Ease of Return to Work
Colleagues' Acceptance
Customers' Acceptance
Spearman's rho
Ease ofReturn to
WorkColleagues'Acceptance
Customers'Acceptance
Correlation is significant at the .01 level (2-tailed).**.
Pearson r vs. Spearman rho
Difference based on whether you were willing to consider rating scale:
Definitely no (1) to (9) Definitely yes
to be interval or ordinal
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 causalityCross-lagged strategy
Time 1Var X (TV violence)
Var Y (AggressiveBehaviors)
Simple Covariation
Covariation and causality
Cross-lagged strategy
Time 1 Time 2
Var X (TV violence) Var X (TV violence)
Var Y (Aggressive Var Y (Aggressive
Behaviors) Behaviors)
Simple Covariation
Covariation and causality
Cross-lagged strategy
Time 1 Time 2
Var X (TV violence) Var X (TV violence)
Var Y (Aggressive Var Y (Aggressive
Behaviors) Behaviors)
Which direction of cause – effect receives stronger support
Y as Cause
X as Cause
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 controlledBecause you simply select for or measure your variables, have less potential to isolate the
variables of interest from other extraneous variables – creating…
“Third” Variable ProblemThe 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
(correlation of the residuals for X and Y after removing relationship with Z) – clearer after
regression
X
Z
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
Simple CovariationCorrelations
1 .699** .341**
. .000 .004
71 71 71
.699** 1 .349**
.000 . .003
71 71 71
.341** .349** 1
.004 .003 .
71 71 71
Pearson Correlation
Sig. (2-tailed)
N
Pearson Correlation
Sig. (2-tailed)
N
Pearson Correlation
Sig. (2-tailed)
N
Investment ModelInvestments Rating
Investment ModelCommitment Rating
How long in months?
InvestmentModel
InvestmentsRating
InvestmentModel
CommitmentRating
How longin months?
Correlation is significant at the 0.01 level (2-tailed).**.
Correlations
1.000 .165
. .171
0 68
.165 1.000
.171 .
68 0
Correlation
Significance (2-tailed)
df
Correlation
Significance (2-tailed)
df
Investment ModelCommitment Rating
How long in months?
Control VariablesInvestment ModelInvestments Rating
InvestmentModel
CommitmentRating
How longin months?
Commitment and How long in months you have been in the relationship are correlated at +.349
When control for Investments made to relationship, correlation reduced to +.165
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