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


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 0.05 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


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


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


Sig. (2-tailed)

N


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



1 two variables covary (covariation)Correlation coefficients can provide a reasonable test of condition #1

Is there evidence for significant (systematic) covariation?


3 other potential causes controlled

Simple 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


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


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





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


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?


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

Documents

Simple Covariation