Inference: Two-Way Tables

Government Spending

• Do people that identify as belonging to different U.S. political parties have different views about government spending?
• We will explore the relationship between party affiliation and opinions on government spending on both national defense and space exploration

Hypotheses

Data:

• gss2016 ¹ dataset
• Subset of General Social Survey (GSS) data from 2016
• party (Dem, Ind, or Rep)
• natarms opinion on current level of government spending on national defense
• natspac opinion on current level of government spending on space exploration
• 149 respondents

Military Spending

• With more than 2 groups, we can’t use a single difference in proportions to compare groups
• We will use the \(X^2\) statistic (chi-squared) to measure the difference between groups

• First compute the expected cell counts, assuming \(H_0\)
• Overall proportion of people that said “too little” spending = 61/149 = 0.4094
• If no association between party and opinion, we expect 0.4094 proportion of dems to have this opinion
• Expected count for dems with opinion “too little” = \(0.4094\cdot 43 = 17.60\)

• Compute \(\frac{(observed\,count-expected\,count)^2}{expected\,count}\) for each cell
• Add values to obtain \(X^2\) statistic, \[\begin{array}{rcl}X^2 &=& 0.02+0.01+0.10 \\ & + & 3.05 + 0.61 + 1.73 \\ & + & 7.30 + 1.02 + 5.13 \\ & =& 18.97\end{array}\]

• Or just ask Jamovi…
• Use the Independent Samples analysis listed under the Frequencies menu
• Drag party into Rows and natarms into Columns

 χ² Tests                              
 ───────────────────────────────────── 
         Value       df    p           
 ───────────────────────────────────── 
   χ²    18.96998     4    0.0007967   
   N          149                      
 ───────────────────────────────────── 

Spending on Space Exploration

 χ² Tests                              
 ───────────────────────────────────── 
         Value       df    p           
 ───────────────────────────────────── 
   χ²    1.326060     4    0.8569388   
   N          149                      
 ───────────────────────────────────── 

Randomization Test for Independence

• We can randomly permute the response (opinion) to simulate the null hypothesis being true
• For each permuted sample, we calculate value of the \(X^2\) statistic
• Let’s construct a null distribution for the military spending question
• We can do this using the Randomize module in Jamovi

Histogram of \(X^2\) statistics for 1,000 random permutations. Observed value indicated by dashed vertical line.

• There were no values of \(X^2\) that were as extreme as the observed value
• The p-value is approximately 0
• We can conclude that opinions on military spending are associated with political party (the two variables are not independent)

Test for Independence Using a Mathematical Model

• \(r\) is the number of rows and \(c\) is the number of columns in the two-way table (no totals)

• Both two-way tables satisfy the large samples condition (at least 5 expected counts in each cell)
• In both cases there are 3 rows and 3 columns in the table, so \(df=(3-1)\times(3-1)=4\)

Chi-squared disribution with \(df=4\). Purple line shows observed value for space question. Red line shows observed value for military question.

• The p-value is the area under the curve that is beyond the observed \(X^2\) value
• We can use the Independent Samples analysis listed under the Frequencies menu in Jamovi to calculate the p-value

• Here are the p-values for the two hypothesis tests

Military Spending

 χ² Tests                              
 ───────────────────────────────────── 
         Value       df    p           
 ───────────────────────────────────── 
   χ²    18.96998     4    0.0007967   
   N          149                      
 ───────────────────────────────────── 

Space Exploration

 χ² Tests                              
 ───────────────────────────────────── 
         Value       df    p           
 ───────────────────────────────────── 
   χ²    1.326060     4    0.8569388   
   N          149                      
 ───────────────────────────────────── 

• As with the randomization-based test, the p-value is very small (<0.001) for the military spending question
• The p-value is quite large (\(p=0.857\)) for the space exploration space exploration question
• We cannot reject null hypothesis in the latter case
• It is plausible that opinion on government spending on space exploration and political affiliation are independent

\(X^2\) distributions for different \(df\)

Chi-squared disributions with different degrees of freedom.

• Chi-squared distribution is more peaked for lower \(df\)
• Thicker tail for higher \(df\)