Setup

# load libraries
library("ggplot2")
library("ggthemes")
library("RColorBrewer")
library("grid")

# load inference code
source("http://bit.ly/dasi_inference")

# load gss data
load(url("http://bit.ly/dasi_gss_data"))

Data

For my analysis I rely on the data set General Social Survey (GSS) (source: http://d396qusza40orc.cloudfront.net/statistics/project/gss.Rdata). The GSS is a sociological survey used to collect data on demographic characteristics and attitudes of residents of the United States.

The survey is conducted face-to-face with an in-person interview by the National Opinion Research Center at the University of Chicago, of adults (18+) in randomly selected households. The survey was conducted every year from 1972 to 1994 (except in 1979, 1981, and 1992). Since 1994, it has been conducted every other year. (source: http://en.wikipedia.org/wiki/General_Social_Survey)

It is an observational study (survey), where respondents are observed and not exposed to a treatment. The respondents are selected randomly. Scope of inference: This data cannot be used for establishing causal links, since there is no random assignment. Scope of interest: The population of interest are residents of the USA. Generalizability is possible, since the respondents were selected randomly. One possible source of bias could be that randomly selected respondents choose not to respond (non-response) so that the sample is not generalizable anymore. The data set contains 57,061 cases (observations). Each case describes a respondent’s interview result, including 114 statements (variables).

The two variables of interest are VIEW ON HOMOSEXUAL RELATIONS (categorical) and HIGHEST YEAR OF SCHOOL COMPLETED (numerical).

Data cleansing

In order to work with the data, I remove all observations that state a NA for the variable VIEW ON HOMOSEXUAL RELATIONS. My assumption is that the distribution of the variable is the same as for the people who have not preferred to answer (24,272 observations). I removed observations with missing values for HIGHEST YEAR OF SCHOOL COMPLETED in the same manner (75 observations).

# delete all NAs for homosex var
gssNA <- gss[rowSums(is.na(gss[95]))==0,]
# delete all NAs for education var
gssNA <- gssNA[rowSums(is.na(gssNA[8]))==0,]

Since I am only interested in the strong opinions on homosexual relations I concentrate on respondents who voted either Always Wrong or Not Wrong At All and I ignore respondents with a not strong opinion, e.g., Sometimes Wrong, Almst Always Wrg or Other.

# Select only the responses of interest
gssNA$strongviewshomosex <- as.character(gssNA$homosex)
gssNA <- gssNA[gssNA$strongviewshomosex != "Sometimes Wrong",]
gssNA <- gssNA[gssNA$strongviewshomosex != "Almst Always Wrg",]
gssNA <- gssNA[gssNA$strongviewshomosex != "Other",]
gssNA$strongviewshomosex <- as.factor(gssNA$strongviewshomosex)

HIGHEST YEAR OF SCHOOL COMPLETED

This section explores the numerical variable HIGHEST YEAR OF SCHOOL COMPLETED (gssNA$educ). The summary for the variable shows a mean of 12.6 years and a median of 12 years. The histogram shows a multimodal distribution. The different degrees very likely explain the peaks (multimodality) in the histogram. The second figure shows histograms differentiated by the variable DEGREE. We also know from the summary of the variable that 25% of all respondents – which answered the same sex relationship question – have less or equal than 11 years of education; and 75% have less or equal than 14 years of education. The maximum lies by 20 years of education.

12 years of education falls into the US high school degree, and hence, I argue that this degree is the most common for US residents. The maximum is 20 years of education, reserved for graduate students.

summary(gssNA$educ)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    0.00   11.00   12.00   12.56   14.00   20.00

sd(gssNA$educ)

## [1] 3.178399

The histogram shows a multimodal distribution that is clearly centered at 12 years of education. The multimodality is explained by the respondents different degrees (another GSS variable).

# Create histogram
educhist <- ggplot(gssNA, aes(x = educ)) + geom_histogram(binwidth = 1) + labs(title = "Histogram of HIGHEST YEAR OF SCHOOL COMPLETED", x = "HIGHEST YEAR OF SCHOOL COMPLETED") 
# Create degree histogram
educdegreehist <- ggplot(gssNA, aes(x = educ)) + geom_histogram(binwidth = 1) + facet_wrap(~degree) + labs(title = "Histogram of HIGHEST YEAR OF SCHOOL COMPLETED, separated by degree", x = "HIGHEST YEAR OF SCHOOL COMPLETED")
# Print both histograms
multiplot(educhist, educdegreehist, cols=2)

Strong OPINION ON HOMOSEXUAL RELATIONS

In this section I explore the categorical variable OPINION ON HOMOSEXUAL RELATIONS with the remaining levels Always Wrong and Not Wrong At All that I refer to as strong opinions. The summary for the variable shows that the majority of the respondents answered that homosexual relations are Always Wrong with 21.542 cases and the minority sees it as Not Wrong At All with 7.271 cases. When we differentiate this by year we see a shift in the distribution of the answers. In 1973 the majority answered with Always Wrong. This changes over time. In 2010 and 2012 the respondents almost equally answered with either Always Wrong or Not Wrong At All.

#by(gssNA$strongviewshomosex, gssNA$year, summary)
summary(gssNA$strongviewshomosex)

##     Always Wrong Not Wrong At All 
##            21542             7271

sd(gssNA$strongviewshomosex)

## [1] 0.4343693

# Create overall barplot
barplotall <- ggplot(gssNA, aes(x = strongviewshomosex)) + geom_bar() + labs(title = "OPINION ON HOMOSEXUAL RELATIONS", y = "Count", x = "OPINION ON HOMOSEXUAL RELATIONS") + coord_flip()
# Create barplot, separated by years
barplotyears <- ggplot(gssNA, aes(x = strongviewshomosex)) + geom_bar() + facet_wrap(~year) + labs(title = "OPINION ON HOMOSEXUAL RELATIONS, differentiated by year", y = "Count", x = "OPINION ON HOMOSEXUAL RELATIONS") + coord_flip()
# Print both histograms
multiplot(barplotall, barplotyears, cols=2)

Set the hypotheses

H0: On average, US citizen that find homosexual relationships Not Wrong At All have the same years of education than US citizen that find homosexual relationships Always Wrong.

HA: On average, US citizen that find homosexual relationships Not Wrong At All have more years of education than US citizen that find homosexual relationships Always Wrong.

Calculate the point estimate and standard error

Firstly, I calculate the standard error. To do this, I need the standard error for each level of categorical variable for the education variable. If the conditions are met, I draw the sampling distribution, shade the p-value, and calculate the test statistic. On the basis of the test statistic I make a decision and interpret it in context of the project question.

#Determine standard error
by(gssNA$educ, gssNA$strongviewshomosex, sd)

## gssNA$strongviewshomosex: Always Wrong
## [1] 3.096514
## -------------------------------------------------------- 
## gssNA$strongviewshomosex: Not Wrong At All
## [1] 2.895745

Calculating the standard error for comparing two means follows this formula:

\[SE = \sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}}\] \[SE = \sqrt{\frac{2.8957^{2}}{7271}+\frac{3.0965^{2}}{21542}} = SE = \sqrt{\frac{8.38507849}{7271}+\frac{9.58831225}{21542}}\] \[SE = \sqrt{0.00115322218264337780222802915692+0.00044509851685080308235075666140563}\]

\[SE = 0.03997900323287438946824542558142\]

Secondly, I calculate the point estimate (mean). To do so, I need to determine the mean for each level of categorical variable for the education variable.

#Determine mean
by(gssNA$educ, gssNA$strongviewshomosex, mean)

## gssNA$strongviewshomosex: Always Wrong
## [1] 12.03811
## -------------------------------------------------------- 
## gssNA$strongviewshomosex: Not Wrong At All
## [1] 14.11993

\[Z = \frac{point estimate - null value}{SE}\] \[Z = \frac{14.1199-12.0381}{0.03997900323287438946824542558142} = 52.072\] I use a significance level of 0.05.

Check conditions

After determining the standard error and the point estimate, I check the conditions in order to apply the inference method.

• Independence within groups: The GSS survey is conducted face-to-face with an in-person interview by the National Opinion Research Center at the University of Chicago, of adults (18+) in randomly selected households. Hence, this condition is met.
• Independence between groups: The observations are not paired data and hence independent. This condition is met.
• Sample size/ skew: Each sample size is larger than 30 observations. \[n_{Always Wrong} = 21542\] and \[n_{Not Wrong At All} = 7271\]. This condition is met.

Draw sampling distribution, shade p-value, calculate test statistic

inference(gssNA$educ, gssNA$strongviewshomosex, est="mean", type="ht", method="theoretical", alternative="greater", null=0, order=c("Not Wrong At All", "Always Wrong"))

## Response variable: numerical, Explanatory variable: categorical
## Difference between two means
## Summary statistics:
## n_Not Wrong At All = 7271, mean_Not Wrong At All = 14.1199, sd_Not Wrong At All = 2.8957
## n_Always Wrong = 21542, mean_Always Wrong = 12.0381, sd_Always Wrong = 3.0965

## Observed difference between means (Not Wrong At All-Always Wrong) = 2.0818
## H0: mu_Not Wrong At All - mu_Always Wrong = 0 
## HA: mu_Not Wrong At All - mu_Always Wrong > 0 
## Standard error = 0.04 
## Test statistic: Z =  52.072 
## p-value =  0