Introduction

We saw from the diagram created by Allen Downey that there is only one test! We will now use this philosophy to develop a technique that allows us to mechanize several Statistical Models in that way, with nearly identical code.

We will use two packages in R, mosaic and the relatively new infer package, to develop our intuition for what are called permutation based statistical tests.

Testing for Two or More Proportions

Let us try a dataset with Qualitative / Categorical data. This is the General Social Survey GSS dataset, and we have people with different levels of Education stating their opinion on the Death Penalty. We want to know if these two Categorical variables have a correlation, i.e. can the opinions in favour of the Death Penalty be explained by the Education level?

Since data is Categorical ( both variables ), we need to take counts in a table, and then implement a chi-square test. In the test, we will permute the Education variable to see if we can see how significant its effect size is.

## 
## categorical variables:  
##             name  class levels    n missing
## 1         Region factor      7 2765       0
## 2         Gender factor      2 2765       0
## 3           Race factor      3 2765       0
## 4      Education factor      5 2760       5
## 5        Marital factor      5 2765       0
## 6       Religion factor     13 2746      19
## 7          Happy factor      3 1369    1396
## 8         Income factor     24 1875     890
## 9       PolParty factor      8 2729      36
## 10      Politics factor      7 1331    1434
## 11     Marijuana factor      2  851    1914
## 12  DeathPenalty factor      2 1308    1457
## 13        OwnGun factor      3  924    1841
## 14        GunLaw factor      2  916    1849
## 15 SpendMilitary factor      3 1324    1441
## 16     SpendEduc factor      3 1343    1422
## 17      SpendEnv factor      3 1322    1443
## 18      SpendSci factor      3 1266    1499
## 19        Pres00 factor      5 1749    1016
## 20      Postlife factor      2 1211    1554
##                                     distribution
## 1  North Central (24.7%) ...                    
## 2  Female (55.6%), Male (44.4%)                 
## 3  White (79.1%), Black (14.8%) ...             
## 4  HS (53.8%), Bachelors (16.1%) ...            
## 5  Married (45.9%), Never Married (25.6%) ...   
## 6  Protestant (53.2%), Catholic (24.5%) ...     
## 7  Pretty happy (57.3%) ...                     
## 8  40000-49999 (9.1%) ...                       
## 9  Ind (19.3%), Not Str Dem (18.9%) ...         
## 10 Moderate (39.2%), Conservative (15.8%) ...   
## 11 Not legal (64%), Legal (36%)                 
## 12 Favor (68.7%), Oppose (31.3%)                
## 13 No (65.5%), Yes (33.5%) ...                  
## 14 Favor (80.5%), Oppose (19.5%)                
## 15 About right (46.5%) ...                      
## 16 Too little (73.9%) ...                       
## 17 Too little (60%) ...                         
## 18 About right (49.7%) ...                      
## 19 Bush (50.6%), Gore (44.7%) ...               
## 20 Yes (80.5%), No (19.5%)                      
## 
## quantitative variables:  
##   name   class min  Q1 median   Q3  max mean       sd    n missing
## 1   ID integer   1 692   1383 2074 2765 1383 798.3311 2765       0

Note how all variables are Categorical !! Education has five levels:

##   Education    n
## 1   Left HS  400
## 2        HS 1485
## 3    Jr Col  202
## 4 Bachelors  443
## 5  Graduate  230
## 6      <NA>    5

##   DeathPenalty    n
## 1        Favor  899
## 2       Oppose  409
## 3         <NA> 1457

Let us drop NA entries in Education and Death Penalty. And set up a table for the chi-square test.

## [1] 1307    2

## # A tibble: 10 × 5
##    Education DeathPenalty count edu_count edu_prop
##    <fct>     <fct>        <int>     <int>    <dbl>
##  1 Bachelors Favor          135       206    0.655
##  2 Bachelors Oppose          71       206    0.345
##  3 Graduate  Favor           64       114    0.561
##  4 Graduate  Oppose          50       114    0.439
##  5 Jr Col    Favor           71        87    0.816
##  6 Jr Col    Oppose          16        87    0.184
##  7 HS        Favor          511       711    0.719
##  8 HS        Oppose         200       711    0.281
##  9 Left HS   Favor          117       189    0.619
## 10 Left HS   Oppose          72       189    0.381

##             Education
## DeathPenalty Left HS  HS Jr Col Bachelors Graduate
##       Favor      117 511     71       135       64
##       Oppose      72 200     16        71       50

## X.squared 
##  23.45093

## 
## 	Pearson's Chi-squared test
## 
## data:  tally(DeathPenalty ~ Education, data = gss2002)
## X-squared = 23.451, df = 4, p-value = 0.0001029

##                X.squared df   p.value                     method alternative
## X-squared...1  1.1268272  4 0.8899928 Pearson's Chi-squared test          NA
## X-squared...2  6.1348278  4 0.1893030 Pearson's Chi-squared test          NA
## X-squared...3 10.1774328  4 0.0375426 Pearson's Chi-squared test          NA
## X-squared...4  3.9776503  4 0.4090390 Pearson's Chi-squared test          NA
## X-squared...5  3.9712187  4 0.4099150 Pearson's Chi-squared test          NA
## X-squared...6  0.3383714  4 0.9872044 Pearson's Chi-squared test          NA
##                                                                   data .row
## X-squared...1 tally(DeathPenalty ~ shuffle(Education), data = gss2002)    1
## X-squared...2 tally(DeathPenalty ~ shuffle(Education), data = gss2002)    1
## X-squared...3 tally(DeathPenalty ~ shuffle(Education), data = gss2002)    1
## X-squared...4 tally(DeathPenalty ~ shuffle(Education), data = gss2002)    1
## X-squared...5 tally(DeathPenalty ~ shuffle(Education), data = gss2002)    1
## X-squared...6 tally(DeathPenalty ~ shuffle(Education), data = gss2002)    1
##               .index
## X-squared...1      1
## X-squared...2      2
## X-squared...3      3
## X-squared...4      4
## X-squared...5      5
## X-squared...6      6

##  prop_TRUE 
## 0.00019998