Question

In: Statistics and Probability

From the 2016 General Social Survey, when we cross-classify political ideology (with 1 being most liberal...

From the 2016 General Social Survey, when we cross-classify political ideology
(with 1 being most liberal and 7 being most conservative) by political party affiliation
for subjects of ages 18–27, we get:
-------------------------------------------------------------
1 2 3 4 5 6 7
Democrat 5 18 19 25 7 7 2
Republican 1 3 1 11 10 11 1
-------------------------------------------------------------
When we use R to model the effect of political ideology on the probability of being
a Democrat, we get the results:
-------------------------------------------------------------
> y <- c(5,18,19,25,7,7,2); n <- c(6,21,20,36,17,18,3)
> x <- c(1,2,3,4,5,6,7)
> fit <- glm(y/n ~ x, family=binomial(link=logit), weights=n)
> summary(fit)
Estimate Std. Error z value Pr(>|z|)
(Intercept) 3.1870 0.7002 4.552 5.33e-06
x -0.5901 0.1564 -3.772 0.000162
---
Null deviance: 24.7983 on 6 degrees of freedom
Residual deviance: 7.7894 on 5 degrees of freedom
Number of Fisher Scoring iterations: 4
> confint(fit)
2.5 % 97.5 %
(Intercept) 1.90180 4.66484
x -0.91587 -0.29832
-------------------------------------------------------------
a. Report the prediction equation and interpret the direction of the estimated effect.
b. Construct the 95% Wald confidence interval for the effect of political ideology.
Interpret and compare to the profile likelihood interval shown.
c. Conduct the Wald test for the effect of x. Report the test statistic, P-value, and
interpret.
d. Conduct the likelihood-ratio test for the effect of x. Report the test statistic, find
the P-value, and interpret.
e. Explain the output about the number of Fisher scoring iterations

Solutions

Expert Solution

a) Prediction equation:

log ( p / (1-p)) = 3.1870 - 0.5901 x

where p = predicted probability of being a Democrat

The predicted value of y = np

Coefficient of x is negative here. A person is more likely to be a Democrat if he is most liberal. There is a negative relationship between the tendency to become democrat and being conservative.

b) 95% Wald confidence interval:

{p-1.96.sqrt (p(1-p)/(N+4)), p-1.96.sqrt (p(1-p)/(N+4))}

p= predicted value of being a democrat

N= sum of n values = total number of experiments = (6 + 21 + 20 + 36 + 17 + 18 + 3) = 121

For each specific class in x, we have different predicted value of p. For each class, we have different confidence interval based on the value of p.

profile likelihood interval shown above is calculated for expected (mean) value of log odds of the probability of being a democrat.

c) Wald test in R:

wald.test(b = coef(fit), Sigma = vcov(fit), Terms = 2)

Test statistic value: 14.2

p-value: 0.00016

d) LR test in R:

lrtest(fit, "x")

Test statistics value: -13.258

p-value: 3.72e-05

e) Fisher scoring iterations is used in R to maximize the likelihood while fitting the model.It is equivalent to iteratively reweighted least squares method. Here 4 iterations are used to get the maximum likelihood estimation of coefficients while fitting Logistic regression.


Related Solutions

Data from 1991 General Social Survey classify a sample of Americans according to their gender and...
Data from 1991 General Social Survey classify a sample of Americans according to their gender and their opinion about afterlife (example from A. Agresti, 1996, “Introduction to categorical data analysis”). The opinions about afterlife were classified into two categories: Yes and No (or undecided). For example, for the females in the sample - 435 said that they believed in an afterlife and 147 said that they did not or were undecided. Gender Belief in Afterlife Yes No or Undecided Females...
6. The following is from the General Social Survey, 2016. Interpret the following frequency distribution and...
6. The following is from the General Social Survey, 2016. Interpret the following frequency distribution and the mode, median, or mean, for the variable CCTV, which asks if the American government should have the right to keep people under video surveillance in public areas. Statistics Statistics CCTV Civil liberties - video surveillance   N Valid 1358 Missing 1509 Mean 2.26 Median 2.00 Mode 2 Std. Deviation 1.008 Variance 1.016 Range 3 CCTV Civil liberties - video surveillance Frequency Valid Percent Cumulative...
QUESTION 1: The University of Chicago's General Social Survey (GSS) is the nation’s most important social...
QUESTION 1: The University of Chicago's General Social Survey (GSS) is the nation’s most important social science sample survey. The GSS asked a random sample of adults their opinion about whether astrology is very scientific, sort of scientific, or not at all scientific. Here is a two-way table of counts for people in the sample who had three levels of higher education degrees: Degree Held Junior College Bachelor Graduate Not at all scientific 45 124 71 Very or sort of...
2. The data below are from the 2016 General Social Survey. Respondents were asked to describe...
2. The data below are from the 2016 General Social Survey. Respondents were asked to describe whether their religion was (1) fundamentalist, (2) moderate or (3) liberal. Respondents were also asked whether they believed in evolution from the statement “Human beings developed from animals.” Respondents not believing in evolution said (1) “false” while supporters of evolution said (2) “true”. Which variable below is the independent variable? The dependent variable? Correctly percentage the data in the table. (Do not calculate the...
. (25.09) The University of Chicago's General Social Survey (GSS) is the nation’s most important social...
. (25.09) The University of Chicago's General Social Survey (GSS) is the nation’s most important social science sample survey. The GSS asked a random sample of adults their opinion about whether astrology is very scientific, sort of scientific, or not at all scientific. Here is a two-way table of counts for people in the sample who had three levels of higher education degrees: Degree Held Junior College Bachelor Graduate Not at all scientific 42 122 73 Very or sort of...
Political parties want to know what groups of people support them. The General Social Survey (GSS)...
Political parties want to know what groups of people support them. The General Social Survey (GSS) asked its 2014 sample, "Generally speaking, do you usually think of yourself as a Republican, Democrat, Independent, or what?" The GSS is essentially an SRS of American adults. Here is a large two-way tale breaking down the responses by the highest degree the subject held: None High School Junior High Bachelor Graduate Strong Democrat 53 198 23 81 64 Not strong Democrat 52 204...
Political parties want to know what groups of people support them. The General Social Survey (GSS)...
Political parties want to know what groups of people support them. The General Social Survey (GSS) asked its 2014 sample, "Generally speaking, do you usually think of yourself as a Republican, Democrat, Independent, or what?" The GSS is essentially an SRS of American adults. Here is a large two-way tale breaking down the responses by the highest degree the subject held: None High School Junior College Bachelor Graduate Strong Democrat 53 198 23 81 64 Not strong Democrat 52 204...
Political parties want to know what groups of people support them. The General Social Survey (GSS)...
Political parties want to know what groups of people support them. The General Social Survey (GSS) asked its 2014 sample, "Generally speaking, do you usually think of yourself as a Republican, Democrat, Independent, or what?" The GSS is essentially an SRS of American adults. Here is a large two-way tale breaking down the responses by the highest degree the subject held: None High School Junior College Bachelor Graduate Strong Democrat 53 198 23 81 64 Not strong Democrat 52 204...
In 2012, the General Social Survey asked a random sample of adults, "Compared to most people,...
In 2012, the General Social Survey asked a random sample of adults, "Compared to most people, how informed are you about politics?" Suppose that the following are the data classified by their responses to this question and their age group (the data has been modified slightly for testing purposes): Not at All A little Somewhat Very Extremely Age 20-29 7 29 28 13 0 Age 30-39 16 29 56 22 8 Age 40-49 3 24 50 26 14 Age 50...
1- From the 2008 General Social Survey, females and males were asked about the number of...
1- From the 2008 General Social Survey, females and males were asked about the number of hours a day that the subject watched TV. Females (n = 698) reported a mean of 3.08 hours with a standard deviation of 2.70 hours. Males (n = 626) reported a mean of 2.87 hours with a standard deviation of 2.61 hours. Test that the mean hours of TV watched by men and women is different from zero at the 5% significance level. n...
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT