Streamlining the Johnson-Neyman Technique for Two-Way Interactions

Transcription

1 Georgia Southern University Digital Southern Georgia Southern University Research Symposium Apr 16th, 9:30 AM - 10:30 AM Streamlining the Johnson-Neyman Technique for Two-Way Interactions W. Georgia Southern University, scarden@georgiasouthern.edu Follow this and additional works at: research_symposium Recommended Citation, W., "Streamlining the Johnson-Neyman Technique for Two-Way Interactions" (2016). Georgia Southern University Research Symposium This presentation (open access) is brought to you for free and open access by the Programs and Conferences at Digital Commons@Georgia Southern. It has been accepted for inclusion in Georgia Southern University Research Symposium by an authorized administrator of Digital Commons@Georgia Southern. For more information, please contact digitalcommons@georgiasouthern.edu.

2 Streamlining the Johnson-Neyman Technique for Two-Way Interactions Work with Nick Holtzmann, Dept. of Psychology Georgia Southern University April 15, 2016

3 A Motivating Example Matt s fraternity is having a party, and Matt s job is ensuring everyone enjoys their drinks responsibly.

4 A Motivating Example Matt knows there is a positive correlation between number of beers (X 1 ) and blood alcohol level (Y ), so he decides to limit everyone to three drinks.

5 A Motivating Example Matt knows there is a positive correlation between number of beers (X 1 ) and blood alcohol level (Y ), so he decides to limit everyone to three drinks. The statisical model he begins with is Y = γ 0 + γ 1 X 1 + ɛ, where ɛ is a zero-mean error term.

6 One Size Does Not Fit All Matt soon realizes that three drinks will not affect all individuals equally.

7 One Size Does Not Fit All Matt soon realizes that three drinks will not affect all individuals equally. Body mass (X 2 ) moderates the effect of X 1 on Y.

8 A model Matt expands his model to: Y = γ 0 + γ 1 X 1 + γ 2 X 2 + γ 3 X 1 X 2 + ɛ, where Y is blood alcohol level, the response variable, X 1 is number of drinks, the focal predictor, X 2 is body mass, the moderator, ɛ is a Normal, zero-mean, uncorrelated error term,

9 A model Matt expands his model to: Y = γ 0 + γ 1 X 1 + γ 2 X 2 + γ 3 X 1 X 2 + ɛ, where Y is blood alcohol level, the response variable, X 1 is number of drinks, the focal predictor, X 2 is body mass, the moderator, ɛ is a Normal, zero-mean, uncorrelated error term, Some insight can be gained by expressing the prediction equation in the form Ŷ = (ˆγ 0 + ˆγ 2 X 2 ) + (ˆγ 1 + ˆγ 3 X 2 )X 1.

10 Rewriting the model Since we are primarily concerned with the effect X 1 has on Y, we can set ˆω 0 (X 2 ) = ˆγ 0 + ˆγ 2 X 2 ˆω 1 (X 2 ) = ˆγ 1 + ˆγ 3 X 2 Simple slope of X 1 and express the prediction equation as Ŷ = ˆω 0 (X 2 ) + ˆω 1 (X 2 )X 1. Question: How does the value X 2 moderate the effect of X 1 on Y?

11 Pick-a-point method Also known as simple slopes analysis or spotlight analysis. Pick a few points for X 2. For these fixed points, determine whether ˆω 1 (x 2 ) is significantly different from zero.

12 Pick-a-point method Also known as simple slopes analysis or spotlight analysis. Pick a few points for X 2. For these fixed points, determine whether ˆω 1 (x 2 ) is significantly different from zero. For fixed X 2 = x 2, the standard error of ˆω 1 (x 2 ) = ˆγ 1 + ˆγ 3 x 2 is Var(ˆγ 1 ) + 2x 2 Cov(ˆγ 1, ˆγ 3 ) + x2 2Var(ˆγ 3). SEˆω1 (x 2 ) = The test statistic is t = ˆω 1(x 2 ) SEˆω1 (x 2 )

13 Simple Slopes Picture

14 Drawbacks Cons: 1 Fixed values of X 2 are ultimately arbitrary. 2 Blind to significance of ˆω 1 (x 2 ) at all other values of X 2.

15 Johnson-Neyman Technique Idea: Run pick-a-point in reverse. Set the test statistic equal to the critical value t α/2 and solve for x 2. ˆγ 1 + ˆγ 3 x 2 = t α/2. Var(ˆγ 1 ) + 2x 2 Cov(ˆγ 1, ˆγ 3 ) + x2 2Var(ˆγ 3)

16 Johnson-Neyman Technique Idea: Run pick-a-point in reverse. Set the test statistic equal to the critical value t α/2 and solve for x 2. ˆγ 1 + ˆγ 3 x 2 = t α/2. Var(ˆγ 1 ) + 2x 2 Cov(ˆγ 1, ˆγ 3 ) + x2 2Var(ˆγ 3) This results in a quadratic in X 2 : ax bx 2 + c = 0 with a = t 2 α/2 Var (ˆγ 3) ˆγ 2 3, b = 2t 2 α/2 Cov(ˆγ 1, ˆγ 3 ) 2ˆγ 1ˆγ 3 c = t 2 α/2 Var(ˆγ 1) ˆγ 2 1. The solutions, x2 and x 2, define the regions of X 2 for which the effect of X 1 on Y is significant.

17 Three Possibilities Case 1: Two real roots are produced, but only one, x 2, is within the range of measurements of X 2. Then either X 1 is significant when X 2 < x 2. X 1 is significant when X 2 > x 2.

18 Case 1 Picture

19 Three Possibilities Case 2: Two real roots are produced, and both are within the range of measurements of X 2. Then either Subcase 2a: X 1 is significant when x 2 < X 2 < x 2. Subcase 2b: X 1 is significant when x 2 < X 2 and X 2 < x 2.

20 Case 2a Picture

21 Case 2b Picture

22 Three Possibilities Case 3: Two complex roots are produced, or two real roots outside the range of measurements of X 2 are produced. Then either Subcase 3a: X 1 is significant for all values within the range of X 2. Subcase 3b: X 1 is not significant for any values within the range of X 2.

23 Case 3a Picture

24 Case 3b Picture

25 Previous Implementations SAS and SPSS - no native implementation. The PROCESS add-on (Hayes, 2013) adds functionality, but requires additional coding and image editing for a publication-ready graph. R - the rockchalk and probemod packages produce poor or mediocre quality graphs. Linear model and creation of testslopes object must be pre-computed.

26 probemod package output

27 Our Implementation Nick and I have been working on in Excel with Visual Basic. Our goals are free availability, accessibility to users without programming skills, and high-quality, publication ready visuals.

28 In Conclusion Limitations Maximum of 1000 observations. Restricted to two explanatory variables. For next version Add error-catching sheet (singular data matrix, leverage values of 1, etc.) Add instructions for exporting figure in various formats.

29 References Hayes, Andrew F. Introduction to Mediation, Moderation, and Conditional Process Analysis. The Guilford Press Johnson, P. O., and Neyman, J. Tests of certain linear hyoptheses and their applications to some educational problems. Statistical Research Memoirs MacKinnon, James G. and White, Halbert. Some Heteroskedasticity-Consistent Covariance Matrix Estimators with Improved Finite Sample Properties. Journal of Econometrics White, Halbert. A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity. Econometrica

30 The End Any questions?

31

32