Multiple Regression

Multiple regression is an effective statistical tool that is used to predict a variable given a set of at least two quantitative independent variables. The aim of this tool is to determine the relationship between several predictor variables and a dependent variable. Ideally, it assumes that the variables are linearly related which can be revealed using a scatter line plot; the variables are quantitatively measured and are normally distributed, homoscedasticity of the residual variance for the predictors’ levels, and lack of multicollinearity between the predictor variables (Howell, 2011). This paper seeks to establish the relationship between weight, anxiety, and systolic pressure. 

Section 1: Data Description 

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The data for this week entails three variables systolic pressure, weight, and anxiety. The analysis focuses on identifying is there is a linear relationship between the three variables – in that how well weight and anxiety predict systolic pressure. It indicates that, both anxiety and weight are the predictor variables or independent variables (IV) while the systolic pressure is the dependent variable (DV). Based on the model the factors are depicted as systolic pressure (Y), anxiety as X1 and Weight as X2. In this case, the three variables are measured using a ratio scale. The sample size is 25 participants (n = 25). 

Section 2: Testing Assumptions 

Normality test

Figure 1: Histogram for Anxiety (X1)

Anxiety data distribution is normally distributed as seen in the above histogram and normality curve. It is clear that most of the data is distributed within the mean (19.12, SD = 6.955). The data is approximately symmetrical indicating normality of data. 

Figure 2: Histogram for Weight (X2)

Most of the scores for the weight are seen to be distributed or spread around the mean (167.56, SD = 68.34) indicating that the variable is normally distributed. 

Figure 3: Histogram for Systolic Pressure (Y)

The systolic pressure is normally distributed as seen in the above histogram and the frequency curve. Majority of the data is distributed within the mean (147.52, SD = 20.38). Considering the three histograms, it can be concluded that the three variables meet the normality assumption of a multiple regression analysis. 

Linearity test 

Figure 4: Scatter plots 

Considering the above Scatterplot matrix, a pairwise bivariate linear relationship can be identified. Considering (X1,X2) it can be seen that there are not extreme bivariate outliers while there is a linear relationship between the two variables. Still, it can be seen that there is a linear relationship between X1 and Y while a pairwise analysis of (X1, Y) shows no notable bivariate outliers. Lastly, the weight and systolic pressure shows a linear relationship with no bivariate outliers. It can be concluded that, the variables have a linear relationship which is one of the basic assumptions of multiple regression.  

Multicollinearity 

Figure 5: Zero-order correlations

Correlations
Control VariablesAnxietyWeightSBP
-none-aAnxietyCorrelation1.000.017.281
Significance (2-tailed)..936.173
df02323
WeightCorrelation.0171.000.453
Significance (2-tailed).936..023
df23023
SBPCorrelation.281.4531.000
Significance (2-tailed).173.023.
df23230
SBPAnxietyCorrelation1.000-.129
Significance (2-tailed)..548
df022
WeightCorrelation-.1291.000
Significance (2-tailed).548.
df220
a. Cells contain zero-order (Pearson) correlations.

           The multicollinearity assumes that there is no linear relationship between the predictor variables or that there is no strong correlation between the independent variables (Howitt and Cramer, 2011). In this case, anxiety and weight have a very weak positive correlations though it is not statistically significant at 0.05 (Pearson r = 0.017, P = 0.936). Therefore, this tells that the multicollinearity assumption is met for the multiple regression analysis. 

Homoscedasticity 

The homoscedasticity assumption in multiple regression analysis claims that the deviance between the actual and predicted scores of the dependent variable (Y) is relatively equal or uniform across all the levels of the independent variables (Howell, 2011). 

Considering the above scatter plots for the residual values of Y against X1 and X2, it is clear that the deviance of the residuals is equal or common in the two plots which prove the assumption of homoscedasticity is met – this proves constant variance of the standardized residuals from the fitted values. The spread of the residuals along the regression line appears the same for the two independent variables. Therefore, the assumption of homoscedasticity is not violated and the multiple regression can now be analyzed since all the assumptions are met. 

Section 3: Research question, hypothesis and alpha

Overall model research question and hypothesis statements 

  1. Do weight and anxiety predict systolic pressure?

Based on the above research question, the study hypothesis can be stated as follows:

Null hypothesis: Anxiety and weight do not significantly predict variance in systolic pressure

Alternative hypothesis: Anxiety and weight significantly predict variance in systolic pressure 

H0: R = 0

H1: R ≠ 0

Predictor research questions and hypothesis statement 

  1. Does weight predict variance in systolic pressure?
  2. Does anxiety predict variance in systolic pressure?

Hypothesis statement for X1

H0: b1 = 0

H1: b1 ≠ 0

Hypothesis statement for X2

H0: b2 = 0

H1: b2 ≠ 0 

Alpha

The analysis is based on a 5% significance level or alpha – which depicts the probability of making a type I error. 

Section 4: Findings and Model output

The assumptions of the multiple regressions were met based on the findings in section 3 above. Accordingly, there is a linear relationship between the variables, there is lack of multicollinearity between anxiety and weight, and variables are normally distributed, while the standardized residuals of the systolic pressure revealed equal variances across all the levels of the predictors. Therefore, this makes the multiple regression suitable in testing the hypothesis. 

Table 1: Regression Model output

Model Summaryb
ModelRR SquareAdjusted R SquareStd. Error of the Estimate
1.529a.280.21518.056
a. Predictors: (Constant), Weight, Anxiety
b. Dependent Variable: SBP

The R is given as 0.529 and this shows the square root of the R-squared and typically shows the correlation between the predicted and observed values of the DV (Systolic pressure). Accordingly, there is a medium correlation between the observed and predicted values of systolic pressure (R = 0.529). On the other hand, R-squared determines the proportion of the model variance explained by the predictors. For instance, anxiety and weight explained 28.0% of the total variation in systolic pressure. This is the effect size of the model as explained by the predictors. 

Table 2: ANOVA output

ANOVAa
ModelSum of SquaresdfMean SquareFSig.
1Regression2791.84121395.9214.282.027b
Residual7172.39922326.018
Total9964.24024
a. Dependent Variable: SBP
b. Predictors: (Constant), Weight, Anxiety

Anxiety and weight significantly predict variation in the systolic pressure, F (2, 22) = 4.282, P = 0.027. In this case, the null hypothesis for the regression model (R = 0) is rejected since p < 0.05 and concluded that both the anxiety and weight significantly predict for any changes or variations in the systolic pressure. 

Table 3: Coefficients ouput

Coefficientsa
ModelUnstandardized CoefficientsStandardized CoefficientstSig.
BStd. ErrorBeta
1(Constant)113.21012.4209.115.000
Anxiety.623.412.2741.512.145
Weight.134.054.4492.479.021
a. Dependent Variable: SBP

    The constant of the model (variation of the model not influenced by the predictors) was 113.210 was significant at alpha 0.05, t = 9.115, P = 0.000. Anxiety has a beta of 0.623 that was not a good predictor of systolic pressure, t = 1.512, p = 0.145. This tells that the b1 = 0 indicating that the null hypothesis for anxiety was retained. In other words, a unit change in anxiety had no impact on systolic pressure. Such an aspect may be shown by the partial correlations where anxiety and systolic pressure had a weak positive correlation that was not significant, R = 0.281, p = 0.173.

Weight significantly predicts systolic pressure, t = 2.479, p = 0.021. Such a p –value tells that the null hypothesis for weight was rejected. Therefore, it can be concluded that weight is a significant predictor of systolic pressure. Additionally, the relationship between weight and systolic pressure can be explained by the partial relationship between the two, R = 0 .453, p = 0.023. An increase in weight by one unit (such as pound) increases systolic pressure by 0.134 units. 

Section 5: Discussion and conclusions

In conclusion, personal weight is seen to influence systolic pressure based on the study findings. Anxiety is not a good predictor of systolic pressure and it cannot be included in the model for making predictions. In other words, the regression model can be re-written as Y = 113.210 + 0.134*X2 where 113.210 is the model constant, 0.134 is the beta associated with weight, and X2 is the weight now. The model excludes anxiety as it is seen as an insignificant predictor in this analysis. 

Strengths of multiple regression 

  • It can estimate the effect size of a linear relationship for more than two predictors on a single dependent variable 
  • It has the ability to identify outliers in a distribution and their effect on the variance of the dependent variable 

Limitations 

  • Incomplete data may give false findings
  • It needs justification of assumptions 
  • Falsely using correlation to indicate causation 
  • May be complex in analysis – incase the assumption tests are not carried properly, it may lead to flawed conclusions.

References

Howell, D. C. (2011). Statistical Methods for Psychology. Wadsworth Publishing.

Howitt, D., & Cramer, D. (2011). Introduction to Research Methods in Psychology. (3rd ed.). (pp. 164, 179-181). Harlow, Essex: Pearson Education Limited

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