Continuous Assessment III

on

Regression Analysis

 

 

Table of Contents

Introduction ………………………………………………………………………………………………………..3

1. Standard Multiple Regression……………………………………….3

2. Logistic Regression………………………………………………..8

3. References………………………………………………………9

INTRODUCTION

This report depicts the result and analysis of two tests performed on two different datasets in order to carry out the regression analysis. In simple words, regression is used to find out the probability of one variable using one or more variables. Regression is basically of two types i.e. linear regression and logistic regression. If a single variable is used to estimate or predict the occurring of a variable, it is called simple linear regression. Moreover, if two or more than two variables are used to estimate the occurring of a variable, it is termed as multiple linear regression. The two tests performed here are the examples of multiple linear regression and logistic regression respectively. Both the tests are carried out using the IBM SPSS Statistics Data Editor Software. The output generated by performing the tests are discussed in the following sections of the paper.

 

1. Standard Multiple Regression

In Multiple Regression, there are three types of models, i.e. Standard or Simultaneous multiple regression, Hierarchical or Sequential multiple regression and Step-wise multiple regression. In this study, the Standard multiple regression test is used. Here, the value of the independent variable is predicted based on the values of the two or more independent variables. In this study four independent variables are used. The primary goal of this analysis is to determine to what extent of the variance in dependent variable can be elucidated by the independent variables. The data set used for this test is taken from the United Kingdom Forest Condition Survey (Data.gov.uk, no date). This test is performed in the IBM SPSS software (Pallant, 2009)for the research considering the below details:

Research Question:

•         To what extent do the independent variables (local crown density, DBH , tree dominance and crown form in pine) can predict the dependent variable (absolute crown density) ?

The dependent and the independent variables:

•         Independent variables – local crown density, DBH , tree dominance and crown form in pine

•         Dependent variable –  absolute crown density

The assumptions for this test are as follows:

•         The dependent variable should be continuous.

•         According to the formula, N > 50+8m, the sample size should be more than 74 as four independent variables are taken into consideration.

•         The relationship between independent variables should not be multicollinear or singular.

•         All the variables in the test should not have very high or very low scores (outliners).

•         The residuals are checked whether they are distributed normally, whether they have linearity and homoscedasticity.  

The output of the test is provided and explained as follows:

Figure 1: Table showing the mean, standard deviation and sample size of each variable.

Figure 2: Table showing the correlation between dependent and the independent variables.

Figure 3: Table showing the R squared value.

Figure 4: Table showing the ANOVA test.

Figure 5: Table showing the regression coefficients.

Figure 6: Table showing the collinearity diagnostics.

Figure 7: Table showing the residual statistics of all the variables.

Figure 8: Figure showing zresid Normal P-P Plot.

Figure 9: Figure showing the scatter Plot.

The analysis of the output from Standard Multiple Regression:

•         The table in figure 1 shows the descriptive statistics of all the variables of the test such as the mean, standard deviation and the sample size. The key parameter to check whether the sample size follows the formula N > 50+8m i.e., the sample size should be greater than 74. The table clearly shows that the sample is more than 74.

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•         The table in figure 2 determines the correlations between the independent variables and the dependent variables. The important point to be noticed in this table is whether the independent variables shows at least some relationship with the dependent variable. According to the Pearson Correlations, it should be more than 0.3. Here, the correlation between the absolute crown density and the local crown density is 0.883, which is more than 0.3. Also, the correlation between all the independent variables is less than 0.7 which meets the required condition.

•         The table in the figure 3 shows the R squared value, which is 0.794. The R squared value determines the total variance of the dependent variable.

•         In figure 4, the the significance value in ANOVA should be less than 0.05 and the significance value in our test is 0.00 which is less than 0.0005. Therefore, this condition is met.

•         Figure 5 shows that the tolerance for all the independent variables is greater than 0.10. It is also observed that the Variance Inflation Factor (VIF) for all the independent variables is less than the recommended value, 10; which clearly implies that there is no collinearity.

•         The plot in figure 8 shows that the values are reasonably along the line, suggesting a slight deviation and scatter plot is distributed in a rectangular fashion showing that they’re arranged in the shape of rectangular.

•         From our model we can say that our independent variables are influencing by 81.2 % on dependent variable. And Number of admissions is contributing the most.

•         Therefore, this study clearly elucidates that the independent variable is influencing by 79.4 % on dependent variable; and the local crown density is contributing the most.

Logistic regression refers to the prediction of a categorial variable using two or more categorial variables. The variable that is being predicted can either be quantitative or qualitative. Logistics regression is further divided into two categories i.e. binomial logistic regression and multinomial logistic regression. If the dependent variable is dichotomous (two possible values) then it is referred as binary logistic regression whereas, if there are more than two categories, it is termed as multinomial logistic regression.

For performing this test, we used a dataset depicting the registered raw milk premises in England, Wales and Northern Ireland. This test is performed using the IBM SPSS Statistics software and output generated is presented and explained below.

Research Question:

•         What factors can be responsible to predict that the users will provide a good compliance rating to the raw milk?

The dependent and the independent variables:

•         Independent variables – Number of branches (defines high availability of the product), is the milk provided is cow’s milk or not (coded 1 as yes and 0 as no).

•         Dependent variable –  Compliance rating (coded 1 as good and 0 as satisfactory)

The assumptions for this test are as follows:

When it comes to logistic regression, it is said that assumptions are not made on the basis of scores of the predictor or independent variables. But the correlation between the independent variables does affect the assumptions.

Case Processing Summary

Unweighted Casesa

N

Percent

Selected Cases

Included in Analysis

169

100.0

Missing Cases

0

.0

Total

169

100.0

Unselected Cases

0

.0

Total

169

100.0

a. If weight is in effect, see classification table for the total number of cases.

In the first table, i.e. the case processing summary table, the first thing one must check is that all the expected number of cases are present in the table or not and there is no missing case. In this table, N indicates the sample size which is 169.

Dependent Variable Encoding

Original Value

Internal Value

Satisfactory

0

Good

1

The second table depicts the coding pattern for the dependent variable. In our case, the dependent variable is the compliance rating coded as 0 for satisfactory and 1 for good. Thus, in the data if 0 is encountered it means the raw milk has a satisfactory rating whereas if 1 is encountered it is assumed to have a good rating.

Categorical Variables Codings

Frequency

Parameter coding

(1)

(2)

Number of Branches

3

11

.000

.000

4

103

1.000

.000

5

55

.000

1.000

Cow’s Milk

no

28

.000

yes

141

1.000

The above table depicts the coding pattern of the independent or predictor variables. Here, the number of branches of the milk premises is not coded, so it simply signifies the numeric value. Coming to the second predictor variable i.e. cow’s milk, it is coded as 0 for no and 1 for yes. The second column of the table named as frequency refers to the number of cases involved in the test.

 

 

Block 0: Beginning Block

Block 0 depicts the output of the analysis without considering the independent or predictor variables.

Iteration Historya,b,c

Iteration

-2 Log likelihood

Coefficients

Constant

Step 0

1

170.639

1.195

2

169.691

1.370

3

169.689

1.379

4

169.689

1.379

a. Constant is included in the model.

b. Initial -2 Log Likelihood: 169.689

c. Estimation terminated at iteration number 4 because parameter estimates changed by less than .001.

Classification Tablea,b

Observed

Predicted

ComplianceRating

Percentage Correct

Satisfactory

Good

Step 0

ComplianceRating

Satisfactory

0

34

.0

Good

0

135

100.0

Overall Percentage

79.9

a. Constant is included in the model.

b. The cut value is .500

In the above provided classification table, the overall percentage correct value is 79.9%.  It indicates that the raw milk is mostly rated as good. This value is evaluated excluding the predictor variables.

Variables in the Equation

B

S.E.

Wald

df

Sig.

Exp(B)

Step 0

Constant

1.379

.192

51.642

1

.000

3.971

Variables not in the Equation

Score

df

Sig.

Step 0

Variables

Number of Branches

1.287

2

.525

Number of Branches(1)

.081

1

.776

Number of Branches(2)

.628

1

.428

Cow’s Milk(1)

.036

1

.850

Overall Statistics

1.332

3

.721

 

Block 1: Method = Enter

Here in block 1, the independent or predictor variables are used, thus, this is the exact logistic regression model analysis.

Iteration Historya,b,c,d

Iteration

-2 Log likelihood

Coefficients

Constant

Number of Branches(1)

Number of Branches(2)

Cow’s Milk(1)

Step 1

1

169.511

1.585

-.423

-.587

.071

2

168.228

2.089

-.767

-1.003

.105

3

168.208

2.217

-.888

-1.131

.109

4

168.208

2.224

-.895

-1.138

.109

5

168.208

2.224

-.895

-1.138

.109

a. Method: Enter

b. Constant is included in the model.

c. Initial -2 Log Likelihood: 169.689

d. Estimation terminated at iteration number 5 because parameter estimates changed by less than .001.

Omnibus Tests of Model Coefficients

Chi-square

df

Sig.

Step 1

Step

1.481

3

.687

Block

1.481

3

.687

Model

1.481

3

.687

The omnibus test shows the overall performance of the regression model. The significant value for ideal case should be less than 0.05, but here the significant value is 0.687. Hence, the predictor variables are said to be less significant as compared to block 0 experiment. Apart from this, the omnibus table also shows the chi square value which is 1.481 with 3 degrees of freedom.

Model Summary

Step

-2 Log likelihood

Cox & Snell R Square

Nagelkerke R Square

1

168.208a

.009

.014

a. Estimation terminated at iteration number 5 because parameter estimates changed by less than .001.

The above table model summary indicates the importance of the regression model. Here the two R square values i.e. Cox & Snell R square and Nagelkerke R square values indicates the variability explained by the model.  Here, these values are 0..9 and 0.14 this means that there is 9% and 14 % variability explained by the model.

Hosmer and Lemeshow Test

Step

Chi-square

df

Sig.

1

.033

3

.998

The Hosmer and Lemeshow test is considered to an important test in the SPSS. For an ideal case, the significant value shown in the Hosmer and Lemeshow test should be greater than 0.05. In our case, the significant value is 0.998 which is greater than 0.05. Thus, our regression model is supported by this test. Apart from this, it also provides the chi square value which is 0.033 with 3 degrees of freedom.

Contingency Table for Hosmer and Lemeshow Test

ComplianceRating = Satisfactory

ComplianceRating = Good

Total

Observed

Expected

Observed

Expected

Step 1

1

3

2.777

8

8.223

11

2

10

10.223

34

33.777

44

3

3

2.930

11

11.070

14

4

17

17.070

72

71.930

89

5

1

1.000

10

10.000

11

Classification Tablea

Observed

Predicted

ComplianceRating

Percentage Correct

Satisfactory

Good

Step 1

ComplianceRating

Satisfactory

0

34

.0

Good

0

135

100.0

Overall Percentage

79.9

The cut value is .500

The classification table evaluates how well can the model predict that the raw milk will get a good compliance rating or satisfactory compliance rating.  Here, the overall percentage correct value is 79.9%. This means that most of the users will provide a good compliance rating. When compared this value to the percentage correct value of Block 0, we find that it is exactly similar.

Variables in the Equation

B

S.E.

Wald

df

Sig.

Exp(B)

95% C.I.for EXP(B)

Lower

Upper

Step 1a

Number of Branches

1.220

2

.543

Number of Branches(1)

-.895

1.081

.686

1

.408

.409

.049

3.398

Number of Branches(2)

-1.138

1.097

1.077

1

.299

.320

.037

2.749

Cow’s Milk(1)

.109

.512

.046

1

.831

1.116

.409

3.044

Constant

2.224

1.110

4.014

1

.045

9.244

a. Variable(s) entered on step 1: Number of Branches, Cow’s Milk.

The above given table provides the importance of each independent variable or predictor variables used in the model. The column wald indicates the values of wald test for each independent variable. The degrees of freedom is 1 for all the variables except number of branches of milk premises. The significant value for ideal case ought to be less than 0.05. In our case, the constant has a significant value 0.045 which is less than 0.05. This is the most significant variable among the independent variables. The B values can either be positive or negative. Here, we have both positive and negative B values. Hence, the negative B value indicates that, users who doesn’t consider the milk to be good won’t give a satisfactory rating. this indicates that the users which have provided a good compliance rating genuinely believe that the milk is good in quality and has a greater availability.

Casewise Listb

Case

Selected Statusa

Observed

Predicted

Predicted Group

Temporary Variable

ComplianceRating

Resid

ZResid

SResid

70

S

S**

.912

G

-.912

-3.211

-2.310

a. S = Selected, U = Unselected cases, and ** = Misclassified cases.

b. Cases with studentized residuals greater than 2.000 are listed.

The above table casewise list provides the ZResid value. This value indicates whether the cases included in the test fit well or not.

References

 

SPSS Survival Manual; A step by step guide to data analysis using SPSS for Windows (Version 12) written by JULIE PALLANT.

https://data.gov.uk/dataset/cccef1ac-fcd2-456a-89eb-49f7206e9ce1/forest-condition-survey-1987-2006

https://data.gov.uk/dataset/f6706084-9c82-4a50-a781-41e0e6229948/raw-drinking-milk-premises-in-england-wales-and-northern-ireland/datafile/6305f564-fcc0-4bd9-9520-b26150d8ce46/preview