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# Logistic Regression Analysis

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A brief introduction to Logistic Regression Analysis, its assumptions, and its application.

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### Logistic Regression Analysis

1. 1. Logistic Regression Analysis -By PIE TUTORS …your statistical partner… www.pietutors.com
2. 2. OUTLINE • Introduction • Assumptions • Model development • Example • References
3. 3. Introduction • Logistic Regression is a statistical method for analyzing a dataset in which there are one or more independent variables that determine an outcome. The outcome is measured with a dichotomous variable, where there are only two possible outcomes. • The goal of logistic regression is to find the best fitting model to describe the relationship between the dichotomous characteristic of interest, and a set of independent variables. • Logistic Regression generates the coefficients of a formula to predict a Logit Transformation of the probability of presence of the characteristic of interest.
4. 4. Assumptions • Assumes a linear relationship between the logit of the IVs and DVs. • Absence of multi-collinearity. • Normal distribution is not assumed for the dependent variable as well as for errors. • Larger samples are needed than for linear regression. • The dependent variable must be a dichotomy (2 categories). • The independent variables need not be interval, nor normally distributed, nor of equal variance within each group.
5. 5. Model Development 1. Binary Logistic Regression As Logistic Regression gives the formula to predict a logit transformation of probability of presence of character of interest, so, the model is, +…….+ In logistic regression, the dependent variable is in fact a logit, which is a log of odds, 1
6. 6. So, the required probability is-
7. 7. 2. Multinomial Logistic Regression Multinomial logit regression is used when the dependent variable in question is nominal and for which there are more than two categories. Two additional assumptions:1. The multinomial logit model assumes that data are case specific, that is, each independent variable has a single value for each case. 2. There is no need for the independent variables to be statistically independent from each other.
8. 8. Model:In multinomial logistic regression there are more than two categories for dependent variable, so the probability of belonging to category ‘j’ is given by- =j)= ∑
9. 9. Example Description:- Entering high school students make program choices among general program, vocational program and academic program. Their choice might be modeled using their writing score and their social economic status. Description of the data:- The data set contains variables on 200 students. The outcome variable is prog, program type. The predictor variables are social economic status, ses, a three-level categorical variable and writing score, write, a continuous variable.
10. 10. Descriptive Statistics Types of program N Mean Std. Deviation General 45 51.33 9.398 Academic 105 56.26 7.943 Vocation 50 46.76 9.319
11. 11. Now, by using multinomial logit modelFitting-criteria Likelihood ratio test model -2 log likelihood Chi-square Intercept only 206.756 Sig. 6 .000 254.986 Final df 48.230
12. 12. Results • The Pseudo R- square value for the model is 0.21. • The likelihood ratio chi-square of 48.23 with a p-value < 0.0001 tells us that our model as a whole fits significantly better than an empty model. And the parameters are corresponding to two equations:= + 1 + 2 + = + 1 + 2 +
13. 13. Parameters Prog. type Wald df Sig. Intercept  1.689 1.896 1 .169 Write  ‐.058 7.320 1 .007 .944 [ses=1] 1.163 5.114 1 .024 3.199 [ses=2] .630 1.833 1 .176 1.877 [ses=3] General B Exp(B) 0 0 Intercept  12.361 1 .000 Write  ‐.114 26.139 1 .000 .893 [ses=1] Vocation  4.236 .983 2.722 1 .099 2.672 [ses=2] 1.274 6.214 1 .013 3.575 [ses=3] 0 0
14. 14. Interpretation • A one-unit increase in the variable write is associated with a .058 decrease in the relative log odds of being in general program versus academic program . • A one-unit increase in the variable write is associated with a .1136 decrease in the relative log odds of being in vocation program versus academic program. • The relative log odds of being in general program versus in academic program will increase by 1.163 if moving from the highest level of ses (ses = 3) to the lowest level of ses (ses = 1).
15. 15. References 1. http://www.schatz.sju.edu/multivar/guide/Logistic.pdf 2. http://www.ats.ucla.edu/stat/spss/dae/mlogit.htm