Overview of Techniques Case 1 Independent Variable is Groups, or Conditions Dependent Variable is continuous ( ) X One sample: Z-test or t-test Two samples: T-test (independent or paired) Three samples: One-way ANOVA F-test Factorial design: Two-Way ANOVA F-test
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Overview of Techniques Case 1 Independent Variable is Groups, or Conditions Dependent Variable is continuous ( ) One sample: Z-test or t-test Two samples:
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Overview of Techniques
Case 1 Independent Variable is Groups, or Conditions
Dependent Variable is continuous ( )X
One sample: Z-test or t-test
Two samples: T-test (independent or paired)
Three samples: One-way ANOVA F-test
Factorial design: Two-Way ANOVA F-test
Overview of Techniques
Case 2 Independent Variable is continuous (x)
Dependent Variable is continuous (y)
One DV, one predictor: correlation, simple linear regression
One DV, multiple predictors: partial correlation, multiple correlation, multiple regression
What if we have two predictor variables?
We want to predict depression. We have measured stress and loneliness.
We can ask several questions:
1) which is the stronger predictor?
2) how well do they predict depression together?
3) what is the effect of loneliness on depression, controlling for stress?
What if we have two predictor variables?
Regressing depression on stress
Predictor Unstandardized Coefficient
Standard error
Standardized Coefficient
t sig
Stress .50 .16 .25 3.125 <.05
Regressing depression on loneliness
Predictor Unstandardized Coefficient
Standard error
Standardized Coefficient
t sig
Loneliness .80 .28 .20 2.86 <.05
R2 = .0625
R2 = .04
Which is the better predictor?How well do they predict depression together?
What is the effect of loneliness controlling for stress?
Pearson’s R2 for loneliness: (a) + (b)
Pearson’s R2 for stress: (c) + (b)
Partial R2 for loneliness: (a)
Partial R2 for stress: (b)
Multiple Regression
Types of effects
Total effect of stress: (b) + (c)
depression
loneliness stress
(a)(b)
(c)
Shared effect of stress and loneliness: (b)
Unique effect of stress: (c)
Slope coefficients in simple regression capture total effects
Slope coefficients in multiple regression capture unique effects
Reasons for Multiple Regression
1) It allows you to directly compare the effect sizes for different predictor variables
2) Adding additional predictors that are related to your Y variable (we call them covariates) allows you to explain more of the residual variance. This makes MS error smaller and increases your power.
2) If you are worried that your key predictor is confounded with other variables, you can “partial them out” or “control for them” in your multiple regression by including them in the analysis.
depression
loneliness stress
(a)(b)
(c)
Two separate regressions
Regressing depression on stress
Predictor Unstandardized Coefficient
Standard error
Standardized Coefficient
t sig
Stress .50 .16 .25 3.125 <.05
Regressing depression on loneliness
Predictor Unstandardized Coefficient
Standard error
Standardized Coefficient
t sig
Loneliness .80 .28 .20 2.86 <.05
R2 = .0625
R2 = .04
Regressing depression on loneliness and stress
Predictor Unstandardized Partial Coefficient
Standard error
Standardized Partial Coefficient
t sig
Intercept 1.8 1.1 - 1.64 .08
Stress .34 .11 .17 3.09 <.05
Loneliness .10 .05 .05 2.00 .06
Multiple R2 = .0625
df = n – p - 12211' XbXbaY
LonelyStressY 10.34.8.1'
A multiple regression
Source SS df s2
Model
Error
Total
2resid
2model
s
sF
YY '
'YY
YY
2)'( YY
2)'( YY
2)( YY 1n
1 pn
p 2models
2resids
2Ys
2Y
2model
s
sR multiple 2
The F-test is for the whole model, doesn’t tell you about individual predictors
Multiple Regression ANOVA
Categorical Predictors in Multiple Regression
Regressing depression on gender (0=female, 1=male)
A dichotomous 0/1 predictor
Gender depression
0 8
1 4
1 10
0 15
1 8
0 14
Regressing depression on gender (0=female, 1=male)
Predictor Unstandardized Partial Coefficient
Standard error
Standardized Partial Coefficient
t sig
Intercept 12.33 1.99 - 6.2 <.01
Gender -5.00 2.81 .66 -1.8 .15
A dichotomous 0/1 predictor
The intercept coefficient tells you the mean depression of the 0 (female) groupThe gender coefficient tells you what to add to get the mean depression of the 1 (male) group
If the gender coefficient is significant, the groups significantly differ
Categorical and Continuous Predictors
in Multiple Regression
Combining Types of Predictors
T-tests and ANOVAs use group variables to predict continuous outcomes
Correlations and simple regressions use continuous variables to predict continuous outcomes
Multiple regressions allow you to use 1) information about group membership and 2) information about other continuous measurements, in the same analysis
Combining Types of Predictors
WHY would we want this?
Imagine that we have a control group and a highly-provoked group, and we also measure the “TypeA-ness” of each participant.
We noticed that because of streaky random sampling, we got more TypeA people in the control group than in the provoked group.
Multiple regression allows us to see if there was an effect of our manipulation, controlling for individual differences in TypeA-ness.
Basically, it allows us to put a situational manipulation and a personality scale measurement into the same study.
Group Provoke TypeA aggression
Control 0 3 3
Control 0 6 4
Control 0 10 6
Control 0 8 5
High 1 2 9
High 1 4 10
High 1 11 24
High 1 12 20
Predictor Unstandardized Partial Coefficient
Standard error
Standardized Partial Coefficient
t sig
Intercept -3.26 2.23 - -1.46 .20
Provoke 10.675 1.89 .734 5.65 <.01
Type A 1.15 .26 .565 4.35 <.01
There is a significant effect of experimental condition and a significant effect of TypeA-ness
General Linear Model
General Linear Model
All of the techniques we’ve covered so far can be expressed as special cases of multiple regression
If you run a multiple regression with an intercept and no slope, the t-test for the intercept is the same as a single sample t-test.
If you put in a dichotomous (0/1) predictor, the t-test for your slope will be the same as an independent samples t-test.
If you put in dummy variables for multiple groups, your regression ANOVA will be the same as your one-way ANOVA or two-way ANOVA.
If you put in one continuous predictor, your β will be the same as your r.
General Linear Model
Plus multiple regression can do so much more!
Looking at several continuous predictors together in one model.
Controlling for confounds.
Using covariates to “soak up” residual variance.
Looking at categorical and continuous predictors together in one model.
Looking at interactions between categorical and continuous variables.