Multiple Regression Sample Paper

QUANTITATIVE RESEARCH METHODS SAMPLE OF MULTIPLE REGRESSION PROCEDURE Prepared by Michael Ling If you are interested to have a copy of the case study, please contact me. 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 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 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 - .6292) = .334, which is a medium 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 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: ,[object Object]

http://www.victoria.ac.nz/psyc/paul-jose-files/modgraph/modgraph.php Appendix ,[object Object],Figure 1: Normal P-P Plot Figure 2: Scatterplot Table 4: ANOVA (Interaction Model)bModelSum of SquaresdfMean SquareFSig.1Regression17298.24435766.08140.819.000aResidual5085.35636141.260Total22383.60039a. Predictors: (Constant), temp_humidity, temperature, humidityb. Dependent Variable: salesModelCollinearity StatisticsToleranceVIF1(Constant)temperature.9541.048humidity.9541.048 Table 5: Model Summary (Interaction Model)bModelRR SquareAdjusted R SquareStd. Error of the Estimate1.879a.773.75411.885a. Predictors: (Constant), temp_humidity, temperature, humidityb. Dependent Variable: sales Table 6: Coefficients (Interaction Model)aModelUnstandardized CoefficientsStandardized CoefficientstSig.95.0% Confidence Interval for BCorrelationsBStd. ErrorBetaLower BoundUpper BoundZero-orderPartialPart1(Constant)257.09660.2974.264.000134.807379.384temperature-6.9762.235-1.413-3.121.004-11.510-2.443.761-.461-.248humidity-76.82517.901-2.322-4.292.000-113.130-40.519.382-.582-.341temp_humidity3.123.6553.6744.770.0001.7954.451.745.622.379a. Dependent Variable: sales Table 7: Model Summary (Interaction Model)ModelRR SquareAdjusted R SquareStd. Error of the EstimateChange StatisticsR Square ChangeF Changedf1df2Sig. F Change1.793a.629.60914.977.62931.397237.0002.879b.773.75411.885.14422.750136.000a. Predictors: (Constant), Zscore(humidity), Zscore(temparature)b. Predictors: (Constant), Zscore(humidity), Zscore(temperature), Zscore(temp_humidity)c. Dependent Variable: salesTable 8: Coefficients (Interaction Model)ModelUnstandardized CoefficientsStandardized CoefficientstSig.95.0% Confidence Interval for BCorrelationsBStd. ErrorBetaLower BoundUpper BoundZero-orderPartialPart1(Constant)96.1002.36840.583.00091.302100.898Zscore(temparature)17.0492.456.7126.943.00012.07322.024.761.752.695Zscore(humidity)5.4952.456.2292.238.031.51910.470.382.345.2242(Constant)96.1001.87951.138.00092.28999.911Zscore(temparature)-33.86010.850-1.413-3.121.004-55.864-11.855.761-.461-.248Zscore(humidity)-55.62312.961-2.322-4.292.000-81.909-29.337.382-.582-.341Zscore(temp_humidity)88.02018.4543.6744.770.00050.594125.446.745.622.379a. Dependent Variable: sales 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

Multiple Regression Sample Paper

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