Question

Read in the Kyphosis data file where y is the outcome of interest and x is our lone predictor.

(a) Fit a logistic regression model of the form logit [P(Y= 1)] =α+β1x. Test the effect of x.

(b) Fit a logistic model of the form logit[P(Y= 1)] =α+β1x+β2x2.  Is the squared term significant?

(c) Plot̂π(x) as a function of x for both the models in (a) and (b). Compare.

This is data

x y

12 1

15 1

42 1

52 1

59 1

73 1

82 1

91 1

96 1

105 1

114 1

120 1

121 1

128 1

130 1

139 1

139 1

157 1

1 0

1 0

2 0

8 0

11 0

18 0

22 0

31 0

37 0

61 0

72 0

81 0

97 0

112 0

118 0

127 0

131 0

140 0

151 0

159 0

177 0

206 0

Answer 1

Using Minitab solving the question

Step 1

- 0 SDOG Minitab - Untitled File Edit Data Calc Stat Graph Editor Tools Window Help Assistant HTX Basic Statistics IOASAGO Regression / Fitted Line Plot... fa - 10 % ANOVA Regression DOE Nonlinear regression... STOOL Control Charts Stability Study Quality Tools Session Reliability/Survival Orthogonal Regression... Multivariate Partial Least Squares... 23-02 Time Series Binary Fitted Line Plot... Welcome to Minitab, Tables | Binary Logistic Regression Nonparametrics Ha Ordinal Logistic Regression... Equivalence Tests 66 Nominal Logistic Regression... Power and Sample Size Poisson Regression - Fit Binary Logistic Model... 4 Predict... Fit Binary Logistic Regression Factorial Pld Perform logistic regression on binary Contour Pld responses. Surface Plot Overlaid Contour Plot... Response Optimizer...

Step 2

Binary Logistic Regression C1 X c2 ☺ Response in binary response/frequency format Response: Y Response event: 1 [requency: (optional) Continuous predictors: Categorical predictors: Model... | Select Options... Graphs... Coding... Results... | OK Stepwise... Storage... | Cancel Ilelp

Step 3

Binary Logistic Regression: Y versus X

Link function Logit
Rows used 40

Response Information

Variable	Value	Count
Y	0	18
	1	22
Total		40

regression equation

- x Minitab - Untitled - [Session] ISS File Edit Data Calc Stat Graph Editor Tools Window Help Assistant H%20TIA DE SOOSSO ifre | -- 11 22 W TOOOM Deviance Table Source DF Adj Dev Adj Mean Chi-Square P-value Regression 1 0.5469 0.5469 0.55 0.460 1 0.5469 0.5469 0.55 0.460 Error 38 54.5042 1.4343 Total 39 55.0511 Model Summary Deviance Deviance R-Sq R-Sq(adj) AIC 0.998 0.00% 58.50 Coefficients Term Coef SE Coef VIF Constant -0.573 0.602 0.00430 0.00585 1.00 Odds Ratios for Continuous Predictors x Odds Ratio 1.0043 95% CI (0.9929, 1.0159) Regression Equation P(1) = exp (Y')/(1 + exp (Y')) Y' = -0.573 + 0.00430 x Goodness-of-Fit Tests Test DF Chi-Square P-value Deviance 54.50 0.040 Pearson 39.93 0.384 Hosmer-Lemeshow 8 7.06 0.530 38 38 Current Worksheet: Worksheet 1 1 2 0 2 9 Editable 16:28 23-02-2020 ~ ENG 2

Binary Logistic Regression: Y versus X Method Link function Logit Rows used 40 Response Information Variable Value Count 22 (Event) 18 Total 40 Deviance Table Source DF Adj Dev Adj Mean Chi-Square P-value Regression 1 0.5469 0.5469 0.55 0.460 1 0.54690 .5469 0.55 0.460 Error 38 54.50421.4343 Total 39 55.0511 Model Summary Deviance Deviance R-sq R-Sq (adj) AIC 0.998 0.00% 58.50 Coefficients Term Coef SE Coef VIF Constant 0.573 0.602 X -0.00430 0.00585 1.00 Odds Ratios for Continuous Predictors Odds Ratio 95% CI X 0.9957 (0.9844, 1.0072) Regression Equation P(0) = exp(Y')/(1 + exp (Y')) Y' = 0.573 - 0.00430 x

(c) Answer

Deviance Residual Plots for Y Normal Probability Plot Histogram Percent Frequency 3.0 -1.6 1.6 -1.5 0.0 1.5 Standardized Residual -0.8 0.0 0.8 Standardized Residual Versus Order Standardized Residual -------- 1 5 10 35 40 15 20 25 30 Observation Order