CORRELATIONS: PART II

Overview

Interpreting Correlations: p-values Challenges in Observational Research

Correlations reduced by poor psychometrics (reliability and validity) Combining measures

Individual predictors often weak Multiple regression

Correlation ≠ causation Directionality and 3rd-variable problems Causal inference Advanced topics: Standardized betas (β) , mediation,

moderation Beyond causation: Prediction and description

Interpreting Correlations

Correlation coefficient Magnitude

Clinical significance, real-world significance, public health significance

p-value Probability of observing an association of a particular

magnitude when no real-world relationship exists More simply: Probability the result is due to sampling

error Even more simply: Probability the result is due to

chance p < .05 means statistically significant, trustworthy,

reliable, not due to chance

Statistical Significance

Depends on the observed effect (magnitude of the correlation)

Depends on the sample size

Challenges Encountered in Observational Research

Correlations reduced by poor psychometrics (reliability and validity)

Individual predictors often weak Correlation ≠ causation

Challenges Encountered in Observational Research

Correlations reduced by poor psychometrics (reliability and validity) Use/make better measures (next unit) Combine measures

Individual predictors often weak Multiple regression

Correlation ≠ causation Methods for improving causal inferences Prediction is fun too

Combining Measures

Any given item (or measure or indicator) has error

Can reduce overall error by combining items, measures, indicators

Many different ways Complex: Many varieties of factor analysis Elegant: Summated scale scores (add

them)

This a different statistic than r, but the same rules apply

Summated

Scale Scores

DOESN’T KNOWFACTOR ANALYSIS

STILL DOES HER JOB

Multiple Regression

Single predictors often weak Human behavior is often

multidetermined Can be used to examine how well

several different independent variables combine to predict a singledependent variable of interest When to use this versus

summated scale scores?r

R

One predictor… not bad

Correlations

1 -.262**

.000

300 300

-.262** 1

.000

300 300

Pearson Correlation

Sig. (2-tailed)

N

Pearson Correlation

Sig. (2-tailed)

N

54. Physical Health

35. Fast Food Eating

54. PhysicalHealth

35. FastFood Eating

Correlation is significant at the 0.01 level (2-tailed).**.

Try finding some more predictors…Correlations

1 .445** -.111 -.262** -.290** -.092

.000 .055 .000 .000 .112

300 300 300 300 300 300

.445** 1 -.081 -.311** -.253** -.143*

.000 .160 .000 .000 .013

300 300 300 300 300 300

-.111 -.081 1 .250** .120* -.042

.055 .160 .000 .038 .472

300 300 300 300 300 300

-.262** -.311** .250** 1 .474** .034

.000 .000 .000 .000 .559

300 300 300 300 300 300

-.290** -.253** .120* .474** 1 .116*

.000 .000 .038 .000 .045

300 300 300 300 300 300

-.092 -.143* -.042 .034 .116* 1

.112 .013 .472 .559 .045

300 300 300 300 300 300

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

54. PhysicalHealth

29. Exercise (1)

34. Meat Eating

35. Fast FoodEating

44. PopDrinking

66. StressLevel

54.PhysicalHealth

29.Exercise

(1)

34.Meat

Eating

35.FastFood

Eating44. PopDrinking

66.StressLevel

Correlation is significant at the 0.01 level (2-tailed).**.

Correlation is significant at the 0.05 level (2-tailed).*.

Now put them in a multiple regression…

Variables Entered/Removedb

44. PopDrinking,29.Exercise(1), 35.Fast FoodEating

a

. Enter

Model1

VariablesEntered

VariablesRemoved Method

All requested variables entered.a.

Dependent Variable: 54. Physical Healthb.

Model Summary

.484a .235 .227 1.68442Model1

R R SquareAdjustedR Square

Std. Error ofthe Estimate

Predictors: (Constant), 44. Pop Drinking, 29. Exercise(1), 35. Fast Food Eating

a.

ANOVAb

257.563 3 85.854 30.259 .000a

839.834 296 2.837

1097.397 299

Regression

Residual

Total

Model1

Sum ofSquares df Mean Square F Sig.

Predictors: (Constant), 44. Pop Drinking, 29. Exercise (1), 35. Fast Food Eatinga.

Dependent Variable: 54. Physical Healthb.

Coefficientsa

5.586 .425 13.156 .000

.359 .050 .383 7.108 .000

-.062 .056 -.066 -1.116 .265

-.115 .042 -.162 -2.775 .006

(Constant)

29. Exercise (1)

35. Fast Food Eating

44. Pop Drinking

Model1

B Std. Error

UnstandardizedCoefficients

Beta

StandardizedCoefficients

t Sig.

Dependent Variable: 54. Physical Healtha.

Correlation ≠ Causation

Mantra of Psyc 1000 Directionality problem 3rd-variable problem

AKA ConfoundingEducation

Level

DepressionSymptom Severity

r = -.20

Correlation ≠ Causation

EducationLevel

DepressionSymptom Severity

r = -.20

Correlation ≠ Causation

EducationLevel

DepressionSymptom Severity

r = -.20

Correlation ≠ Causation

EducationLevel

DepressionSymptom Severity

r = -.20

Correlation ≠ Causation

EducationLevel

DepressionSymptom Severity

r = -.20Parental

SES

Correlation ≠ Causation

PotSmoking

Ice CreamEating

r = .20

Causal Inference

Ability to infer (assert) causation exists on a continuum

Requirements for Causation Internal validity: Rule out 3rd variables

(alternative explanations) Temporal precedence

Also helpful Stronger associations Theoretically plausible Corroborating experimental evidence

3rd–Variable Problem

Methodologic Control If worried about a 3rd variable, control for it in your

sample (e.g., if worried about SES, only study doctors)

Measure 3rd Variables Measure potential confounders to show they are

not correlated with the variables you wish to study Statistically Control for 3rd Variables

Easy peasy. Many statistical techniques for doing this (e.g., partial correlations, ANCOVA), but we’ll just use regression

Only works well if the potential confounder was measured well (breast milk example)

Statistical Control in RegressionVariables Entered/Removedb

44. PopDrinking,29.Exercise(1), 35.Fast FoodEating

a

. Enter

Model1

VariablesEntered

VariablesRemoved Method

All requested variables entered.a.

Dependent Variable: 54. Physical Healthb.

Model Summary

.484a .235 .227 1.68442Model1

R R SquareAdjustedR Square

Std. Error ofthe Estimate

Predictors: (Constant), 44. Pop Drinking, 29. Exercise(1), 35. Fast Food Eating

a. ANOVAb

257.563 3 85.854 30.259 .000a

839.834 296 2.837

1097.397 299

Regression

Residual

Total

Model1

Sum ofSquares df Mean Square F Sig.

Predictors: (Constant), 44. Pop Drinking, 29. Exercise (1), 35. Fast Food Eatinga.

Dependent Variable: 54. Physical Healthb. Coefficientsa

5.586 .425 13.156 .000

.359 .050 .383 7.108 .000

-.062 .056 -.066 -1.116 .265

-.115 .042 -.162 -2.775 .006

(Constant)

29. Exercise (1)

35. Fast Food Eating

44. Pop Drinking

Model1

B Std. Error

UnstandardizedCoefficients

Beta

StandardizedCoefficients

t Sig.

Dependent Variable: 54. Physical Healtha.

Statistical Control in Regression Imagine that cigarette smoking across

the lifespan is correlated with physical health at age 60 (r = -.40)

If you were a cigarette company, what third variables might you blame?

Alcohol use, extraversion, income, education level, poor coping skills

Do a multiple regression and find that smoking is still associated with physical health even after controlling for those variables (β = -.37, p < .001)

Temporal Precedence

Cross-sectional vs. longitudinal study Prospective vs. retrospective study

EducationLevel

T2

DepressionSymptom

Severity T2

EducationLevel

T1

DepressionSymptom

Severity T1

Temporal Precedence

EducationLevel

T2

DepressionSymptom

Severity T2

EducationLevel

T1

DepressionSymptom

Severity T1

Temporal Precedence

EducationLevel

T2

DepressionSymptom

Severity T2

EducationLevel

T1

DepressionSymptom

Severity T1

β = .03

β = .21

Education level at T1 predicts Depression at T2, while controlling for Depression at T1. More or less, Education level at T1 predicts changes in depression.

Mediation

Rather than examining how A causes B, focuses on a causal chain: A causing B causing C…

DepressionSymptom

Severity T2

EducationLevel

T1

Child DepressionSymptom Severity

T3

Moderation

Different from mediation Also called “interaction” and “effect

modification” Means that an association varies by

group Relationship between A and B depends

on C

DepressionSymptom

Severity T2

EducationLevel

T1

β = .21

DepressionSymptom

Severity T2

EducationLevel

T1

β = .11

Males Females

Prediction and Description

Observational research (and correlations) are important in their own right, regardless of whether or not associations are causal

Examples Decision-making research Personalized medicine, MMPI, Pandora,

dating Others

Correlations

1 .146* .124* .127* .123* .208** .197** -.129* -.167** -.175*

.136* .100 .111 -.005 .011 .091 .080 .084 -.009 -.151*

.139* .036 .052 .417** -.017 -.309** -.297** -.255** -.083 -.199**

.141* .206** .050 -.184** .057 .380** .329** .198** -.171** -.076

.256** -.046 .142* .333** .079 -.002 -.088 -.174** -.045 -.269**

.170** .127* .008 .068 .148* .176** .144* -.019 .013 -.142*

.125* -.111 -.074 -.070 .048 .053 .030 -.110 .008 .020

.146* 1 .230** -.047 -.013 .135* .142* .082 -.102 .136*

.124* .230** 1 .101 -.065 .103 .032 -.071 -.066 -.069

.127* -.047 .101 1 .093 -.090 -.142* -.258** -.086 -.038

.123* -.013 -.065 .093 1 .088 .075 .049 .028 -.220**

.208** .135* .103 -.090 .088 1 .676** .151** -.062 -.036

.197** .142* .032 -.142* .075 .676** 1 .093 -.007 -.013

-.129* .082 -.071 -.258** .049 .151** .093 1 .047 .006

-.167** -.102 -.066 -.086 .028 -.062 -.007 .047 1 -.108

-.175* .136* -.069 -.038 -.220** -.036 -.013 .006 -.108 1

46. Cell Phone Use

28. Cleanliness

30. Sadness

39. Laughing

41. Crying

43. Tanning

45. Gambling

57. Encouraged to Read

58. Obama as Change

66. Stress Level

72. Wal-Mart Shopping

74. Sociability

75. Extraversion

80. Body Satisfaction

90. Number of Siblings

96. ACT Score

46. CellPhone Use

57.Encouraged

to Read58. Obamaas Change

66. StressLevel

72. Wal-MartShopping 74. Sociability

75.Extraversion

80. BodySatisfaction

90. Numberof Siblings 96. ACT Score

Correlation is significant at the 0.05 level (2-tailed).*.

Correlation is significant at the 0.01 level (2-tailed).**.

Correlations

1 .102 .294** .104 -.049 .157** .120* .191** .194** -.138* -.037 .029 .012 -.136* .070 .084 -.147* -.098 .047 -.028 -.089 -.029

.183** .550** .261** .014 -.072 .437** .467** .419** .514** -.286** -.212** .194** .019 -.193** -.023 .105 -.248** -.048 .044 -.064 -.044 -.110

-.199** -.411** -.291** -.067 .082 -.318** -.366** -.365** -.287** .457** .490** -.140* .036 .133* -.007 .139* .294** .065 -.027 -.022 .088 .018

.294** .183** 1 .137* -.066 .053 .161** .129* .202** -.077 -.275** .125* .099 -.171** .041 .055 -.124* -.044 -.052 .145* -.066 -.122

.157** .396** .053 -.053 -.047 1 .608** .412** .407** -.203** -.098 .076 .047 -.095 -.084 .078 -.207** -.099 .054 -.105 -.046 .049

.191** .416** .129* .026 -.019 .412** .418** 1 .353** -.226** -.152** .091 -.076 -.092 .018 -.035 -.165** -.034 .097 -.037 -.012 -.006

.194** .283** .202** .143* -.033 .407** .360** .353** 1 -.063 -.074 .093 .031 -.091 -.068 .141* -.164** -.060 .015 .076 -.050 -.055

.223** .490** .213** -.020 -.085 .459** .463** .294** .379** -.361** -.295** .173** .014 -.230** -.093 .034 -.284** -.113* .019 -.056 -.052 -.061

.276** .265** .209** .053 -.051 .283** .287** .215** .278** -.057 -.177** -.018 .033 -.141* .013 .070 -.173** -.113 .115* -.059 -.134* .012

-.155** -.302** -.116* -.049 .078 -.224** -.244** -.269** -.220** .382** .318** -.071 .136* .197** .014 .003 .247** .066 -.023 -.009 .010 .101

.161** .086 .024 -.122* -.085 .068 .055 .121* -.066 -.035 -.129* .203** -.082 -.070 .063 -.057 -.160** .108 .009 .035 .012 -.024

.145* .236** .246** .025 -.081 .244** .281** .278** .380** -.058 -.002 .241** .176** -.266** .053 .208** -.171** -.020 .011 -.063 -.093 -.038

.206** .312** .067 .007 -.064 .270** .180** .389** .233** -.174** -.157** .023 -.012 -.194** -.089 .025 -.130* .017 -.001 -.093 -.126* .025

-.236** -.467** -.241** -.053 .068 -.447** -.489** -.423** -.392** .379** .359** -.045 .008 .097 .021 .013 .307** .078 -.069 -.058 .091 .047

.163** .162** .195** -.105 -.328** .136* .160** .065 .010 -.070 -.031 .019 -.036 -.280** .061 .020 -.073 .047 .001 -.028 -.151** .002

-.165** -.362** -.076 .025 .034 -.508** -.430** -.444** -.245** .294** .238** -.064 .091 .080 -.068 .058 .245** .042 -.022 -.022 .077 .043

89. Oly mpic Viewers hip

27. Happines s (1)

30. Sadnes s

33. Sports Partic ipation

36. Lov ed by Others

38. Trus ting (1)

39. Laughing

55. Mental Heal th

56. Parental Relations hipQual i ty61. Moodines s

69. Pol i tic a l Interes t

74. Soc iabi l i ty

78. Agreeablenes s

84. Depres s ion

85. Frui t Eating

86. Lonel ines s

89. Oly mpicViewers hip

32.Satis fac tion

(1)33. Sports

Partic ipation34. MeatEating

35. Fas tFood Eating

36. Lov edby Others

37. Pers onalImportanc e

38. Trus ting(1) 39. Laughing 40. Anx iety 41. Cry ing 42. Boldnes s 43. Tanning

44. PopDrink ing 45. Gambl ing

46. Cel lPhone Us e

47.Somatiz ation

48.Mc Cain-Bus hEquiv alenc e

49.Pers onal i ty

in the Genes50. Male

Superiori ty51. His tory of

Spank ings

52. GlobalWarming

Ac k nowledgement

Correlation is s igni fic ant at the 0.01 lev el (2-ta i led).**.

Corre lation is s igni fic ant at the 0.05 lev el (2-ta i led).*.