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# Multiple Regression worked example (July 2014 updated)

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• low/med/hi categories are automatically generated by Modgraph when you fill in its data entry table. You enter B, mean and SD of the centered scores of temp and humidity (Table 7, 8). To get the interaction effects of humidity, you put temp as main effect var, humidity as moderator; to get the interaction effects of temp, you put the values of humidity as main effect var, temp as moderator.

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• could you please tell me how did you plot the figures in page no:9 of your multiple regression procedure. As the temperature and humidy are continous, how did you arrive the levels like Low, Medium and High for those.

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### Multiple Regression worked example (July 2014 updated)

1. 1. REGRESSION ANALYSIS July 2014 updated Prepared by Michael Ling Page 1 QUANTITATIVE RESEARCH METHODS SAMPLE OF REGRESSION ANALYSIS Prepared by Michael Ling
2. 2. REGRESSION ANALYSIS July 2014 updated Prepared by Michael Ling Page 2 PROBLEM Create a multiple regression model to predict the level of daily ice-cream sales Mr Whippy can ex pect to make, given the daily temperature and humidity. Using the base model (50 marks): • What is the regression model and regression equation? • What interpretation do you make of the findings? • Is the regression model valid? • Is the sample size adequate? Create an interaction term for temperature and humidity: • Is there an interaction effect in the model? • What is the effect size (F 2 ) of the interaction? • What interpretation do you make of the findings? • Show the interaction effect graphically (e.g., using ModGraph) SOLUTION Base Model The regression model is Sales = a + b*temperature + c*humidity + e where Sales is the criterion variable, temperature and humidity are predictor; a is intercept crosses the Sales axis; b and c are regression coefficients; e is an error term. The regression equation is Sales = -24.112 + 3.513*temperature + 7.589*humidity (Table 1). Since R2 =.629, 62.9% of the variance in ice-cream sales can be explained by temperature and humidity (Table 2). Compared to R2 , adjusted R2 provides a less biased estimate (60.9%) of the extent of the relationship between the variables in the population. The ANOVA is significant (F=31.397, df(regression)=2, df(residual)=37, Sig < .001 ) which means that the two predictors collectively account for a statistically significant proportion of the variance in the criterion variable (Table 3). The B weight for temperature is 3.513, which means that, after controlling for humidity, a 1-unit increase in temperature will result in a predicted 3.513 unit increase in ice-cream sales. The B weight for humidity is 7.589, which means that, after controlling for temperature, a 1-unit
3. 3. REGRESSION ANALYSIS July 2014 updated Prepared by Michael Ling Page 3 increase in temperature will result in a predicted 7.589 unit increase in ice-cream sales (Table 1). The standardized coefficient (Beta) for temperature is .712, which means, after controlling for humidity, a 1 standard deviation (SD) increase in temperature will result in a .712 SD increase in ice-cream sales. Similarly, a 1 SD increase in humidity will result in a .229 SD increase in ice- cream sales (Table 1). Temperature can account for a significant proportion of unique variance in ice-cream sales (t=6.943, Sig < .001) (Table 1). Humidity accounts for a significant proportion of unique variance in ice-cream sales (t=2.238, Sig < 0.05) (Table 1). The Pearson’s correlation between temperature and ice-cream sales is r = .761, and that between humidity and ice-cream sales is r = .382 (Table 1). The partial correlation between temperature and ice-cream sales is .752 and that between humidity and ice-cream sales is .345 (Table 1). The part correlation (sr) for temperature is .695, indicating that approximately 48.3% (.6952 ) of the variance in ice-cream sales can be uniquely attributed to temperature (Table 1). Similarly, approximately 5% (.2242 ) of the variance in ice- cream sales can be uniquely attributed to humidity (Table 1). The Variance Inflation Factors (VIF) of temperature and humidity are both 1.048. As they are both close to 1, multicollinearity is not a problem. From the normal P-P plot, the points are clustered tightly along the diagonal and hence the residuals are normally distributed (Figure 1). The absence of any clear patterns in the spread of points in the scatterplot indicates that the assumptions of normality, linearity and homoscedasticity of residuals are met (Figure 2). Using G*Power and setting alpha = .05 (two-tailed), power = 0.8 and 2 predictors, the results of sample sizes are shown in Table A. As there are 40 samples in this dataset, the effect size is approximately .25 and hence samples are adequate to detect a medium-to-large effect. Interaction Model
4. 4. REGRESSION ANALYSIS July 2014 updated Prepared by Michael Ling Page 4 The ANOVA is significant (F=40.819, df(regression)=3, df(residual)=36, Sig < .001) which indicates that the interaction model is statistically significant (Table 4). Since R2 =.773, 77.3% of the variance in ice-cream sales can be explained by the interaction model with the interaction effect, which is14.4% improvement over the base model (Table 5). The regression equation is Sales = 257.096 – 6.976*temperature – 76.825*humidity + 3.123*temperature*humidity (Table 6). Temperature can account for a significant proportion of unique variance in ice-cream sales (t=-3.121, Sig < .005) (Table 6). Humidity accounts for a significant proportion of unique variance in ice-cream sales (t=-4.292, Sig < .001) (Table 6). The interaction variable can account for a significant proportion of unique variance in ice-cream sales (t=4.770, Sig < .001) (Table 6). The partial correlation between temperature and ice- cream sales is -.461 and that between humidity and ice-cream sales is -.582 (Table 6). The part correlation (sr) for temperature is reduced to -.248, indicating that approximately 6.2% (.2482 ) of the variance in ice-cream sales can be uniquely attributed to temperature (Table 6). Approximately 11.6% (.3412 ) of the variance in ice-cream sales can be uniquely attributed to humidity (Table 6), and approximately 14.3% (.3792 ) of the variance in ice-cream sales can be uniquely attributed to the interaction variable (Table 6). The effect size of the interaction (F2) = (.7732 - .6292 ) / (1 - .7732 ) = .502. Since it is greater than .35, the result is a large effect. The use of VIFs to interpret multicollinearity in a regression model that has interaction effects is erroneous with uncentered variables [1]. As a result, the moderating effect is examined by applying ModGraph[2] on centered scores. The centered scores of the interaction model are the zscores (Table 7 and Table 8). Two ModGraphs are plotted where one examines the moderating relationship when temperature is the main effect (Figure 3) and the other examines moderating relationship when humidity is the main effect (Figure 4). Referring to Figure 3, ice-cream sales is directly proportional to temperature only when humidity is high, ice-cream sales is inversely proportional to temperature when humidity is both
5. 5. REGRESSION ANALYSIS July 2014 updated Prepared by Michael Ling Page 5 medium and low. Thus, humidity moderates the relationship between ice-cream sale and temperature. Referring to Figure 4, ice-cream sales is directly proportional to humidity only when temperature is high, ice-cream sales is inversely proportional to humidity when temperature is both medium and low. Thus, temperature moderates the relationship between ice- cream sale and humidity. References: 1. Robinson, C. & Schumacker, R. E. (2009). Interaction Effects: Centering, Variance Inflation Factor, and Interpretation Issues. Multiple Linear Regression Viewpoints, 35 (1), 6-11. 2. http://www.victoria.ac.nz/psyc/paul-jose-files/modgraph/modgraph.php
6. 6. REGRESSION ANALYSIS July 2014 updated Prepared by Michael Ling Page 6 Appendix Table 1: Base Model - Coefficients Model Unstandardized Coefficients Standardized Coefficients t Sig. 95.0% Confidence Interval for B Correlations B Std. Error Beta Lower Bound Upper Bound Zero- order Partial Part 1 (Constant) -24.112 15.933 -1.513 .139 -56.394 8.171 temperature 3.513 .506 .712 6.943 .000 2.488 4.538 .761 .752 .695 humidity 7.589 3.392 .229 2.238 .031 .717 14.461 .382 .345 .224 a. Dependent Variable: sales Model Collinearity Statistics Tolerance VIF 1 (Constant) temperature .954 1.048 humidity .954 1.048 Table 2: Base Model Summaryb Model R R Square Adjusted R Square Std. Error of the Estimate 1 .793a .629 .609 14.977 a. Predictors: (Constant), humidity, temparature b. Dependent Variable: sales Table 3: Base Model - ANOVAb Model Sum of Squares df Mean Square F Sig. 1 Regression 14084.540 2 7042.270 31.397 .000a Residual 8299.060 37 224.299 Total 22383.600 39 a. Predictors: (Constant), humidity, temparature b. Dependent Variable: sales Table A: Results of G*Power Effect Size .35 .25 .15 Sample Size 28 42 66
7. 7. REGRESSION ANALYSIS July 2014 updated Prepared by Michael Ling Page 7 Figure 1: Normal P-P Plot Figure 2: Scatterplot
8. 8. REGRESSION ANALYSIS July 2014 updated Prepared by Michael Ling Page 8 Table 4: ANOVA (Interaction Model)b Model Sum of Squares df Mean Square F Sig. 1 Regression 17298.244 3 5766.081 40.819 .000a Residual 5085.356 36 141.260 Total 22383.600 39 a. Predictors: (Constant), temp_humidity, temperature, humidity b. Dependent Variable: sales Model Collinearity Statistics Tolerance VIF 1 (Constant) temperature .954 1.048 humidity .954 1.048 Table 5: Model Summary (Interaction Model)b Model R R Square Adjusted R Square Std. Error of the Estimate 1 .879a .773 .754 11.885 a. Predictors: (Constant), temp_humidity, temperature, humidity b. Dependent Variable: sales
9. 9. REGRESSION ANALYSIS July 2014 updated Prepared by Michael Ling Page 9 Table 6: Coefficients (Interaction Model)a Model Unstandardized Coefficients Standardized Coefficients t Sig. 95.0% Confidence Interval for B Correlations B Std. Error Beta Lower Bound Upper Bound Zero- order Partial Part 1 (Constant) 257.096 60.297 4.264 .000 134.807 379.384 temperature -6.976 2.235 -1.413 -3.121 .004 -11.510 -2.443 .761 -.461 -.248 humidity -76.825 17.901 -2.322 -4.292 .000 -113.130 -40.519 .382 -.582 -.341 temp_humidity 3.123 .655 3.674 4.770 .000 1.795 4.451 .745 .622 .379 a. Dependent Variable: sales Table 7: Model Summary (Interaction Model) Model R R Square Adjusted R Square Std. Error of the Estimate Change Statistics R Square Change F Change df1 df2 Sig. F Change 1 .793a .629 .609 14.977 .629 31.397 2 37 .000 2 .879b .773 .754 11.885 .144 22.750 1 36 .000 a. Predictors: (Constant), Zscore(humidity), Zscore(temparature) b. Predictors: (Constant), Zscore(humidity), Zscore(temperature), Zscore(temp_humidity) c. Dependent Variable: sales Table 8: Coefficients (Interaction Model) Model Unstandardized Coefficients Standardized Coefficients t Sig. 95.0% Confidence Interval for B Correlations B Std. Error Beta Lower Bound Upper Bound Zero- order Partial Part 1 (Constant) 96.100 2.368 40.583 .000 91.302 100.898 Zscore(temparature) 17.049 2.456 .712 6.943 .000 12.073 22.024 .761 .752 .695 Zscore(humidity) 5.495 2.456 .229 2.238 .031 .519 10.470 .382 .345 .224 2 (Constant) 96.100 1.879 51.138 .000 92.289 99.911 Zscore(temparature) -33.860 10.850 -1.413 -3.121 .004 -55.864 -11.855 .761 -.461 -.248 Zscore(humidity) -55.623 12.961 -2.322 -4.292 .000 -81.909 -29.337 .382 -.582 -.341 Zscore(temp_humidity) 88.020 18.454 3.674 4.770 .000 50.594 125.446 .745 .622 .379 a. Dependent Variable: sales
10. 10. REGRESSION ANALYSIS July 2014 updated Prepared by Michael Ling Page 10 Figure 3: ModGraph 1 – zscore(temp) as main effect, zscore(humidity) as moderator, zscore(temp*humidity) as interaction variable Figure 4: ModGraph 1 – zscore(humidity) as main effect, zscore(temperature) as moderator, zscore(temp*humidity) as interaction variable -50.00 0.00 50.00 100.00 150.00 200.00 250.00 300.00 low med high SaleofIce-cream Temperature Temperature and Humidity Humidity high med low Grade Humidity Temperature and Humidity Temperature high med low