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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).*.