Categorical DataStatistics I — Data ExplorationDeepayan Sarkar

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Motivation

Suppose we have paired bivariate data $(X_{i}, Y_{i}), i = 1, 2, \dots, n$ , where
- $Y$ is a "response" or "dependent" variable, and
- $X$ is a "predictor" or "independent" variable
So far, we have considered the case where $Y$ is numeric, $X$ is either categorical or numeric

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Example: Categorical $X$

Distribution of $Y$

library(NHANES)
bwplot(BPSysAve ~ Race1 | Gender, data = NHANES, subset = Age >= 21)

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Example: Categorical $X$

Summarize (conditional) distribution of $Y$ by median

bwplot(BPSysAve ~ Race1 | Gender, data = NHANES, subset = Age >= 21,
       panel = panel.average, fun = function(x) median(x, na.rm = TRUE), col = "black")

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Example: Categorical $X$

Summarize (conditional) distribution of $Y$ by mean

bwplot(BPSysAve ~ Race1 | Gender, data = NHANES, subset = Age >= 21,
       panel = panel.average, fun = function(x) mean(x, na.rm = TRUE), col = "black")

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Example: Numeric $X$

Distribution of $Y$

xyplot(BPSysAve ~ log(Weight) | Gender, data = NHANES, subset = Age >= 21, grid = TRUE)

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Example: Numeric $X$

Summarize (conditional) distribution of $Y$ by least squares line

xyplot(BPSysAve ~ log(Weight) | Gender, data = NHANES, subset = Age >= 21, grid = TRUE, smooth = "lm")

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Categorical response

The other combinations we can consider:
- $Y$ categorical, $X$ categorical
- $Y$ categorical, $X$ numeric

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Categorical response

The other combinations we can consider:
- $Y$ categorical, $X$ categorical
- $Y$ categorical, $X$ numeric
In this course, we will only consider the first case
In fact, we will typically consider $Y$ to be binary (two categories)

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Contingency tables summarizing categorical data

Categorical data can be summarized by frequency of each combination
This process is known as cross-tabulation
The resulting summary is known as a contingency table

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Example: Race and Gender

head(NHANES[c("Race1", "Gender")], 20)

     Race1 Gender
1    White   male
2    White   male
3    White   male
4    Other   male
5    White female
6    White   male
7    White   male
8    White female
9    White female
10   White female
11   White   male
12   White   male
13   White   male
14   White female
15 Mexican female
16   White   male
17   Black female
18   White   male
19   White   male
20   Other   male

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Example: Race and Gender

Unique combinations

unique(NHANES[c("Race1", "Gender")])

      Race1 Gender
1     White   male
4     Other   male
5     White female
15  Mexican female
17    Black female
24 Hispanic   male
31    Other female
35  Mexican   male
45    Black   male
46 Hispanic female

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Example: Race and Gender

Count of unique combinations

xtabs(~ Race1 + Gender, NHANES)

          Gender
Race1      female male
  Black       614  583
  Hispanic    320  290
  Mexican     452  563
  White      3221 3151
  Other       413  393

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Example: Race and Gender

Count of unique combinations as data frame

xtabs(~ Race1 + Gender, NHANES) |> array2DF()

      Race1 Gender Value
1     Black female   614
2  Hispanic female   320
3   Mexican female   452
4     White female  3221
5     Other female   413
6     Black   male   583
7  Hispanic   male   290
8   Mexican   male   563
9     White   male  3151
10    Other   male   393

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Representing contingency tables

The above examples show two standard methods to represent contingency tables
The data frame representation is more familiar to us
The "table"-like format is more standard in many situations

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Representing contingency tables

The above examples show two standard methods to represent contingency tables
The data frame representation is more familiar to us
The "table"-like format is more standard in many situations
The "table" representation is generalized to "arrays" in R
We can switch between the representations using xtabs() and array2DF()

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Example: UCBAdmissionsAdmission by gender in 6 departments in a US university
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Example: UCBAdmissionsAdmission by gender in 6 departments in a US university
UCBAdmissions

, , Dept = A
          Gender
Admit      Male Female
  Admitted  512     89
  Rejected  313     19
, , Dept = B
          Gender
Admit      Male Female
  Admitted  353     17
  Rejected  207      8
, , Dept = C
          Gender
Admit      Male Female
  Admitted  120    202
  Rejected  205    391
, , Dept = D
          Gender
Admit      Male Female
  Admitted  138    131
  Rejected  279    244
, , Dept = E
          Gender
Admit      Male Female
  Admitted   53     94
  Rejected  138    299
, , Dept = F
          Gender
Admit      Male Female
  Admitted   22     24
  Rejected  351    317
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Example: `UCBAdmissions`

UCBAdmissions[1, , ]

        Dept
Gender     A   B   C   D   E   F
  Male   512 353 120 138  53  22
  Female  89  17 202 131  94  24

UCBAdmissions[, 1, ]

          Dept
Admit        A   B   C   D   E   F
  Admitted 512 353 120 138  53  22
  Rejected 313 207 205 279 138 351

UCBAdmissions[, , 1]

          Gender
Admit      Male Female
  Admitted  512     89
  Rejected  313     19

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Example: UCBAdmissionsUCBAdmissionsDF <- array2DF(UCBAdmissions) # convert to data frame representation
UCBAdmissionsDF

      Admit Gender Dept Value
Admitted   Male    A   512
Rejected   Male    A   313
Admitted Female    A    89
Rejected Female    A    19
Admitted   Male    B   353
Rejected   Male    B   207
Admitted Female    B    17
Rejected Female    B     8
Admitted   Male    C   120
Rejected   Male    C   205
Admitted Female    C   202
Rejected Female    C   391
Admitted   Male    D   138
Rejected   Male    D   279
Admitted Female    D   131
Rejected Female    D   244
Admitted   Male    E    53
Rejected   Male    E   138
Admitted Female    E    94
Rejected Female    E   299
Admitted   Male    F    22
Rejected   Male    F   351
Admitted Female    F    24
Rejected Female    F   317
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Example: `UCBAdmissions`

Admission by gender

(admitByGender <- xtabs(Value ~ Gender + Admit, data = UCBAdmissionsDF))

        Admit
Gender   Admitted Rejected
  Female      557     1278
  Male       1198     1493

apply(UCBAdmissions, c(1, 2), sum) # alternative using apply() function

          Gender
Admit      Male Female
  Admitted 1198    557
  Rejected 1493   1278

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Example: `UCBAdmissions`

Visualization using bar chart

barchart(admitByGender, auto.key = TRUE, stack = FALSE)

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Example: `UCBAdmissions`

Visualization using mosaic plot

library(vcd)
mosaic(admitByGender)

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Example: `UCBAdmissions`

Visualization of three-way table using bar chart

barchart(Gender ~ Value | Dept, data = UCBAdmissionsDF, groups = Admit,
         origin = 0, auto.key = TRUE, stack = FALSE, as.table = TRUE)

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Example: `UCBAdmissions`

Visualization of three-way table using mosaic plot

mosaic(xtabs(Value ~ Gender + Dept + Admit, UCBAdmissionsDF)) # change order

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Example: `UCBAdmissions`

This dataset is a well-known example that illustrates Simpson's Paradox
Ignoring one variable can lead to wrong conclusions about relationship between two other variables

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Summarizing conditional distribution of $Y$

For numeric $Y$ , we can summarize the distribution by center (mean / median)
For categorical $Y$ , the natural summary is proportions in each category

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Summarizing conditional distribution of $Y$

For numeric $Y$ , we can summarize the distribution by center (mean / median)
For categorical $Y$ , the natural summary is proportions in each category
For binary response, this can be sample proportion of any one category

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Summarizing conditional distribution of $Y$

For numeric $Y$ , we can summarize the distribution by center (mean / median)
For categorical $Y$ , the natural summary is proportions in each category
For binary response, this can be sample proportion of any one category
However, it is also common to measure it by the "odds" of one of the categories

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Odds

Let the probability of an event $A$ be $p = P (A)$
Think of $A$ as the event of "success" ("Admitted" in the example above)

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Odds

Let the probability of an event $A$ be $p = P (A)$
Think of $A$ as the event of "success" ("Admitted" in the example above)
The "odds" are defined as the ratio of success probability and failure probability

$odds of A = \frac{P (A)}{P (A^{C})} = \frac{p}{1 - p}$

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Odds

Let the probability of an event $A$ be $p = P (A)$
Think of $A$ as the event of "success" ("Admitted" in the example above)
The "odds" are defined as the ratio of success probability and failure probability

$odds of A = \frac{P (A)}{P (A^{C})} = \frac{p}{1 - p}$

Examples
- For $p = 0.5$ , the odds are 1 ("50-50 odds")
- For $p = 0.9$ , the odds are $0.9 / 0.1 = 9$ ("9 to 1 odds")
- For $p = 0.2$ , the odds are $0.2 / 0.8 = 0.25$ ("odds are 4 to 1 against")

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Odds as a transformation

The idea of odds is closer to how we usually think of probability
Mathematically, we can think of $p \mapsto \frac{p}{1 - p}$ as a transformation
This transformation converts the bounded parameter $p$ to an unbounded parameter

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Odds as a transformation

The idea of odds is closer to how we usually think of probability
Mathematically, we can think of $p \mapsto \frac{p}{1 - p}$ as a transformation
This transformation converts the bounded parameter $p$ to an unbounded parameter
However, the odds are still bounded below by $0$
It is "asymmetric" in the sense that
- for unfavourable odds, $p < 0.5 ⟹ odds \in [0, 1)$
- for favourable odds, $p > 0.5 ⟹ odds \in (1, \infty)$

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Log odds

We often symmetrize this by considering the log-odds $\log \frac{p}{1 - p}$
The value of log-odds can be any real number

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Log odds

We often symmetrize this by considering the log-odds $\log \frac{p}{1 - p}$
The value of log-odds can be any real number
The log-odds transformation is often useful for proportions

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Example: UN data

data(UN, package = "carData")
xyplot(pctUrban ~ log(ppgdp), data = UN, grid = TRUE,
       type = c("p", "smooth"), col.line = "red") + layer(panel.lmline(x, y))

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Example: UN data

logodds <- function(p) log(p / (1-p))
xyplot(logodds(pctUrban/100) ~ log(ppgdp), data = UN, grid = TRUE, subset = pctUrban < 100,
       type = c("p", "smooth"), col.line = "red") + layer(panel.lmline(x, y))

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Odds ratio

Suppose we want to compare success probability in two groups
For example, probability of admission for males vs females

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Odds ratio

Suppose we want to compare success probability in two groups
For example, probability of admission for males vs females
Odds ratio: Compute odds for both groups, take their ratio

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Odds ratio

Suppose we want to compare success probability in two groups
For example, probability of admission for males vs females
Odds ratio: Compute odds for both groups, take their ratio
Interpretation: If $B$ represents predictor (e.g., gender), odds ratio is

$\frac{P (A ∣ B) / P (A^{C} ∣ B)}{P (A ∣ B^{C}) / P (A^{C} ∣ B^{C})}$

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Odds ratio

Suppose we want to compare success probability in two groups
For example, probability of admission for males vs females
Odds ratio: Compute odds for both groups, take their ratio
Interpretation: If $B$ represents predictor (e.g., gender), odds ratio is

$\frac{P (A ∣ B) / P (A^{C} ∣ B)}{P (A ∣ B^{C}) / P (A^{C} ∣ B^{C})}$

Clearly, log of odds ratio is the difference between log odds (zero means no difference)

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Example: `UCBAdmissions` revisited

Odds ratio for all departments combined

oddsratio <- function(x) { # x = 2 x 2 matrix, column 1 = success, column 2 = failure
    (x[1,1] / x[1,2]) / (x[2,1] / x[2,2])
}
admitByGender

        Admit
Gender   Admitted Rejected
  Female      557     1278
  Male       1198     1493

oddsratio(admitByGender)

[1] 0.5431594

log(oddsratio(admitByGender))

[1] -0.6103524

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Example: `UCBAdmissions` revisited

Odds ratio by department

UCBAdmissions[,, "A"]

          Gender
Admit      Male Female
  Admitted  512     89
  Rejected  313     19

Transpose and reorder to get in desired format

t(UCBAdmissions[, c(2, 1), "A"])

        Admit
Gender   Admitted Rejected
  Female       89       19
  Male        512      313

log(oddsratio(t(UCBAdmissions[, c(2, 1), "A"])))

[1] 1.052076

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Example: UCBAdmissions revisitedlog(oddsratio(t(UCBAdmissions[, c(2, 1), "A"])))

[1] 1.052076
log(oddsratio(t(UCBAdmissions[, c(2, 1), "B"])))

[1] 0.2200225
log(oddsratio(t(UCBAdmissions[, c(2, 1), "C"])))

[1] -0.1249216
log(oddsratio(t(UCBAdmissions[, c(2, 1), "D"])))

[1] 0.08198719
log(oddsratio(t(UCBAdmissions[, c(2, 1), "E"])))

[1] -0.200187
log(oddsratio(t(UCBAdmissions[, c(2, 1), "F"])))

[1] 0.1888958
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Example: Hypertension vs Age

nhsub <- subset(NHANES, Age >= 21 & !is.na(BPSysAve) & Race1 == "White" & Gender == "female")
nhsub$HyperTension <- ifelse(nhsub$BPSysAve > 120, "Yes", "No")
HypTable <- xtabs(~ Age + HyperTension, nhsub)
head(HypTable, 20)

    HyperTension
Age  No Yes
  21 18   4
  22 20   1
  23 42   7
  24 31  15
  25 37   2
  26 35   7
  27 36   4
  28 28   5
  29 29   6
  30 43   3
  31 28  11
  32 26   4
  33 35   6
  34 25   4
  35 21   3
  36 28   4
  37 30  11
  38 46   6
  39 38   8
  40 29   6

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Example: Hypertension vs Age

HypProp <- HypTable[, "Yes"] / rowSums(HypTable)
HypOdds <- HypTable[, "Yes"] / HypTable[, "No"]
HypLogOdds <- log(HypOdds)
Age <- as.numeric(rownames(HypTable))

data.frame(Age, HypProp, HypOdds, HypLogOdds) |> head(15)

   Age    HypProp    HypOdds HypLogOdds
21  21 0.18181818 0.22222222 -1.5040774
22  22 0.04761905 0.05000000 -2.9957323
23  23 0.14285714 0.16666667 -1.7917595
24  24 0.32608696 0.48387097 -0.7259370
25  25 0.05128205 0.05405405 -2.9177707
26  26 0.16666667 0.20000000 -1.6094379
27  27 0.10000000 0.11111111 -2.1972246
28  28 0.15151515 0.17857143 -1.7227666
29  29 0.17142857 0.20689655 -1.5755364
30  30 0.06521739 0.06976744 -2.6625878
31  31 0.28205128 0.39285714 -0.9343092
32  32 0.13333333 0.15384615 -1.8718022
33  33 0.14634146 0.17142857 -1.7635886
34  34 0.13793103 0.16000000 -1.8325815
35  35 0.12500000 0.14285714 -1.9459101

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Example: Hypertension vs Age

xyplot(HypProp ~ Age, grid = TRUE)

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Example: Hypertension vs Age

xyplot(HypOdds ~ Age, grid = TRUE)

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Example: Hypertension vs Age

xyplot(HypLogOdds ~ Age, grid = TRUE)

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Example: Titanic dataTitanic

, , Age = Child, Survived = No
      Sex
Class  Male Female
  1st     0      0
  2nd     0      0
  3rd    35     17
  Crew    0      0
, , Age = Adult, Survived = No
      Sex
Class  Male Female
  1st   118      4
  2nd   154     13
  3rd   387     89
  Crew  670      3
, , Age = Child, Survived = Yes
      Sex
Class  Male Female
  1st     5      1
  2nd    11     13
  3rd    13     14
  Crew    0      0
, , Age = Adult, Survived = Yes
      Sex
Class  Male Female
  1st    57    140
  2nd    14     80
  3rd    75     76
  Crew  192     20
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Example: Titanic dataarray2DF(Titanic)

   Class    Sex   Age Survived Value
  1st   Male Child       No     0
  2nd   Male Child       No     0
  3rd   Male Child       No    35
 Crew   Male Child       No     0
  1st Female Child       No     0
  2nd Female Child       No     0
  3rd Female Child       No    17
 Crew Female Child       No     0
  1st   Male Adult       No   118
 2nd   Male Adult       No   154
 3rd   Male Adult       No   387
Crew   Male Adult       No   670
 1st Female Adult       No     4
 2nd Female Adult       No    13
 3rd Female Adult       No    89
Crew Female Adult       No     3
 1st   Male Child      Yes     5
 2nd   Male Child      Yes    11
 3rd   Male Child      Yes    13
Crew   Male Child      Yes     0
 1st Female Child      Yes     1
 2nd Female Child      Yes    13
 3rd Female Child      Yes    14
Crew Female Child      Yes     0
 1st   Male Adult      Yes    57
 2nd   Male Adult      Yes    14
 3rd   Male Adult      Yes    75
Crew   Male Adult      Yes   192
 1st Female Adult      Yes   140
 2nd Female Adult      Yes    80
 3rd Female Adult      Yes    76
Crew Female Adult      Yes    20
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Example: `Titanic` data

Data on survival status of Titanic passengers by Age / Sex / Class
Exercise: Compute and compare odds of survival for various passenger subgroups

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Are two attributes independent?

Natural next question: Are two attributes independent? (in the population)
We can consider the admissions data as an example

t(UCBAdmissions[, c(2, 1), "C"])

        Admit
Gender   Admitted Rejected
  Female      202      391
  Male        120      205

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Are two attributes independent?

Natural next question: Are two attributes independent? (in the population)
We can consider the admissions data as an example

t(UCBAdmissions[, c(2, 1), "C"])

        Admit
Gender   Admitted Rejected
  Female      202      391
  Male        120      205

Question (about population)
- Assume that this represents a random sample from all potential applicants
- Is that data consistent with what we would expect if there is no gender bias?

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Are two attributes independent?

Alternate formulation: We can calculate the log odds ratio

log(oddsratio(t(UCBAdmissions[, c(2, 1), "C"])))

[1] -0.1249216

Is this value sufficiently different from zero?

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Are two attributes independent?

Alternate formulation: We can calculate the log odds ratio

log(oddsratio(t(UCBAdmissions[, c(2, 1), "C"])))

[1] -0.1249216

Is this value sufficiently different from zero?
To answer this question, we need to know what the odds ratio could have been under independence

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Are two attributes independent?

Alternate formulation: We can calculate the log odds ratio

log(oddsratio(t(UCBAdmissions[, c(2, 1), "C"])))

[1] -0.1249216

Is this value sufficiently different from zero?
To answer this question, we need to know what the odds ratio could have been under independence
Answering such questions is an important reason for studying Statistics formally
This sort of insight is more valuable than simple data summarization and visualization

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Are two attributes independent?

Alternate formulation: We can calculate the log odds ratio

log(oddsratio(t(UCBAdmissions[, c(2, 1), "C"])))

[1] -0.1249216

Is this value sufficiently different from zero?
To answer this question, we need to know what the odds ratio could have been under independence
Answering such questions is an important reason for studying Statistics formally
This sort of insight is more valuable than simple data summarization and visualization
This area of Statistics is known as inference (what we have been doing is descriptive statistics)

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Are two attributes independent?

How can we approach this problem?

X <- t(UCBAdmissions[, c(2, 1), "C"])
X

        Admit
Gender   Admitted Rejected
  Female      202      391
  Male        120      205

log(oddsratio(X))

[1] -0.1249216

Intuitively, bias against female candidates $⟹$ large negative value for log odds ratio
But how large should the value be to decide that there is bias?

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Are two attributes independent?

How can we approach this problem?

X <- t(UCBAdmissions[, c(2, 1), "C"])
X

        Admit
Gender   Admitted Rejected
  Female      202      391
  Male        120      205

log(oddsratio(X))

[1] -0.1249216

Intuitively, bias against female candidates $⟹$ large negative value for log odds ratio
But how large should the value be to decide that there is bias?
We will study such questions using parametric models in the next semester
Here, we will consider a simple intuitive approach

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Are two attributes independent?

We want to study the "distribution" of log odds ratio if there was no bias
Of course, we do not know whether there was any bias

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Are two attributes independent?

We want to study the "distribution" of log odds ratio if there was no bias
Of course, we do not know whether there was any bias
Imagine doing a hypothetical experiment
- Suppose gender of candidate is hidden from admission committee
- This will guarantee that there was no gender bias

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Are two attributes independent?

We want to study the "distribution" of log odds ratio if there was no bias
Of course, we do not know whether there was any bias
Imagine doing a hypothetical experiment
- Suppose gender of candidate is hidden from admission committee
- This will guarantee that there was no gender bias
If the process was indeed gender-neutral, then the outcomes would not have changed

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Are two attributes independent?

We want to study the "distribution" of log odds ratio if there was no bias
Of course, we do not know whether there was any bias
Imagine doing a hypothetical experiment
- Suppose gender of candidate is hidden from admission committee
- This will guarantee that there was no gender bias
If the process was indeed gender-neutral, then the outcomes would not have changed
Even if we added fake random values for gender, the outcomes would not have changed

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Are two attributes independent?

We want to study the "distribution" of log odds ratio if there was no bias
Of course, we do not know whether there was any bias
Imagine doing a hypothetical experiment
- Suppose gender of candidate is hidden from admission committee
- This will guarantee that there was no gender bias
If the process was indeed gender-neutral, then the outcomes would not have changed
Even if we added fake random values for gender, the outcomes would not have changed
This is something we can do easily:
- Take all candidates, keep admission status unchanged, but assign gender randomly
- How many females vs males? Easiest to keep the totals unchanged
- This is equivalent to randomly permuting the gender variable

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Permutation test for independence

To do this, we need data in expanded form

        Admit
Gender   Admitted Rejected
  Female      202      391
  Male        120      205

d <- array2DF(X)
d

  Gender    Admit Value
1 Female Admitted   202
2   Male Admitted   120
3 Female Rejected   391
4   Male Rejected   205

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Permutation test for independencedlong <- d[rep(1:4, d$Value) , c("Gender", "Admit")]
dlong

      Gender    Admit
   Female Admitted
1   Female Admitted
2   Female Admitted
3   Female Admitted
4   Female Admitted
5   Female Admitted
6   Female Admitted
7   Female Admitted
8   Female Admitted
9   Female Admitted
10  Female Admitted
11  Female Admitted
12  Female Admitted
13  Female Admitted
14  Female Admitted
15  Female Admitted
16  Female Admitted
17  Female Admitted
18  Female Admitted
19  Female Admitted
20  Female Admitted
21  Female Admitted
22  Female Admitted
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25  Female Admitted
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36  Female Admitted
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38  Female Admitted
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40  Female Admitted
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42  Female Admitted
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100 Female Admitted
101 Female Admitted
102 Female Admitted
103 Female Admitted
104 Female Admitted
105 Female Admitted
106 Female Admitted
107 Female Admitted
108 Female Admitted
109 Female Admitted
110 Female Admitted
111 Female Admitted
112 Female Admitted
113 Female Admitted
114 Female Admitted
115 Female Admitted
116 Female Admitted
117 Female Admitted
118 Female Admitted
119 Female Admitted
120 Female Admitted
121 Female Admitted
122 Female Admitted
123 Female Admitted
124 Female Admitted
125 Female Admitted
126 Female Admitted
127 Female Admitted
128 Female Admitted
129 Female Admitted
130 Female Admitted
131 Female Admitted
132 Female Admitted
133 Female Admitted
134 Female Admitted
135 Female Admitted
136 Female Admitted
137 Female Admitted
138 Female Admitted
139 Female Admitted
140 Female Admitted
141 Female Admitted
142 Female Admitted
143 Female Admitted
144 Female Admitted
145 Female Admitted
146 Female Admitted
147 Female Admitted
148 Female Admitted
149 Female Admitted
150 Female Admitted
151 Female Admitted
152 Female Admitted
153 Female Admitted
154 Female Admitted
155 Female Admitted
156 Female Admitted
157 Female Admitted
158 Female Admitted
159 Female Admitted
160 Female Admitted
161 Female Admitted
162 Female Admitted
163 Female Admitted
164 Female Admitted
165 Female Admitted
166 Female Admitted
167 Female Admitted
168 Female Admitted
169 Female Admitted
170 Female Admitted
171 Female Admitted
172 Female Admitted
173 Female Admitted
174 Female Admitted
175 Female Admitted
176 Female Admitted
177 Female Admitted
178 Female Admitted
179 Female Admitted
180 Female Admitted
181 Female Admitted
182 Female Admitted
183 Female Admitted
184 Female Admitted
185 Female Admitted
186 Female Admitted
187 Female Admitted
188 Female Admitted
189 Female Admitted
190 Female Admitted
191 Female Admitted
192 Female Admitted
193 Female Admitted
194 Female Admitted
195 Female Admitted
196 Female Admitted
197 Female Admitted
198 Female Admitted
199 Female Admitted
200 Female Admitted
201 Female Admitted
     Male Admitted
1     Male Admitted
2     Male Admitted
3     Male Admitted
4     Male Admitted
5     Male Admitted
6     Male Admitted
7     Male Admitted
8     Male Admitted
9     Male Admitted
10    Male Admitted
11    Male Admitted
12    Male Admitted
13    Male Admitted
14    Male Admitted
15    Male Admitted
16    Male Admitted
17    Male Admitted
18    Male Admitted
19    Male Admitted
20    Male Admitted
21    Male Admitted
22    Male Admitted
23    Male Admitted
24    Male Admitted
25    Male Admitted
26    Male Admitted
27    Male Admitted
28    Male Admitted
29    Male Admitted
30    Male Admitted
31    Male Admitted
32    Male Admitted
33    Male Admitted
34    Male Admitted
35    Male Admitted
36    Male Admitted
37    Male Admitted
38    Male Admitted
39    Male Admitted
40    Male Admitted
41    Male Admitted
42    Male Admitted
43    Male Admitted
44    Male Admitted
45    Male Admitted
46    Male Admitted
47    Male Admitted
48    Male Admitted
49    Male Admitted
50    Male Admitted
51    Male Admitted
52    Male Admitted
53    Male Admitted
54    Male Admitted
55    Male Admitted
56    Male Admitted
57    Male Admitted
58    Male Admitted
59    Male Admitted
60    Male Admitted
61    Male Admitted
62    Male Admitted
63    Male Admitted
64    Male Admitted
65    Male Admitted
66    Male Admitted
67    Male Admitted
68    Male Admitted
69    Male Admitted
70    Male Admitted
71    Male Admitted
72    Male Admitted
73    Male Admitted
74    Male Admitted
75    Male Admitted
76    Male Admitted
77    Male Admitted
78    Male Admitted
79    Male Admitted
80    Male Admitted
81    Male Admitted
82    Male Admitted
83    Male Admitted
84    Male Admitted
85    Male Admitted
86    Male Admitted
87    Male Admitted
88    Male Admitted
89    Male Admitted
90    Male Admitted
91    Male Admitted
92    Male Admitted
93    Male Admitted
94    Male Admitted
95    Male Admitted
96    Male Admitted
97    Male Admitted
98    Male Admitted
99    Male Admitted
100   Male Admitted
101   Male Admitted
102   Male Admitted
103   Male Admitted
104   Male Admitted
105   Male Admitted
106   Male Admitted
107   Male Admitted
108   Male Admitted
109   Male Admitted
110   Male Admitted
111   Male Admitted
112   Male Admitted
113   Male Admitted
114   Male Admitted
115   Male Admitted
116   Male Admitted
117   Male Admitted
118   Male Admitted
119   Male Admitted
   Female Rejected
1   Female Rejected
2   Female Rejected
3   Female Rejected
4   Female Rejected
5   Female Rejected
6   Female Rejected
7   Female Rejected
8   Female Rejected
9   Female Rejected
10  Female Rejected
11  Female Rejected
12  Female Rejected
13  Female Rejected
14  Female Rejected
15  Female Rejected
16  Female Rejected
17  Female Rejected
18  Female Rejected
19  Female Rejected
20  Female Rejected
21  Female Rejected
22  Female Rejected
23  Female Rejected
24  Female Rejected
25  Female Rejected
26  Female Rejected
27  Female Rejected
28  Female Rejected
29  Female Rejected
30  Female Rejected
31  Female Rejected
32  Female Rejected
33  Female Rejected
34  Female Rejected
35  Female Rejected
36  Female Rejected
37  Female Rejected
38  Female Rejected
39  Female Rejected
40  Female Rejected
41  Female Rejected
42  Female Rejected
43  Female Rejected
44  Female Rejected
45  Female Rejected
46  Female Rejected
47  Female Rejected
48  Female Rejected
49  Female Rejected
50  Female Rejected
51  Female Rejected
52  Female Rejected
53  Female Rejected
54  Female Rejected
55  Female Rejected
56  Female Rejected
57  Female Rejected
58  Female Rejected
59  Female Rejected
60  Female Rejected
61  Female Rejected
62  Female Rejected
63  Female Rejected
64  Female Rejected
65  Female Rejected
66  Female Rejected
67  Female Rejected
68  Female Rejected
69  Female Rejected
70  Female Rejected
71  Female Rejected
72  Female Rejected
73  Female Rejected
74  Female Rejected
75  Female Rejected
76  Female Rejected
77  Female Rejected
78  Female Rejected
79  Female Rejected
80  Female Rejected
81  Female Rejected
82  Female Rejected
83  Female Rejected
84  Female Rejected
85  Female Rejected
86  Female Rejected
87  Female Rejected
88  Female Rejected
89  Female Rejected
90  Female Rejected
91  Female Rejected
92  Female Rejected
93  Female Rejected
94  Female Rejected
95  Female Rejected
96  Female Rejected
97  Female Rejected
98  Female Rejected
99  Female Rejected
100 Female Rejected
101 Female Rejected
102 Female Rejected
103 Female Rejected
104 Female Rejected
105 Female Rejected
106 Female Rejected
107 Female Rejected
108 Female Rejected
109 Female Rejected
110 Female Rejected
111 Female Rejected
112 Female Rejected
113 Female Rejected
114 Female Rejected
115 Female Rejected
116 Female Rejected
117 Female Rejected
118 Female Rejected
119 Female Rejected
120 Female Rejected
121 Female Rejected
122 Female Rejected
123 Female Rejected
124 Female Rejected
125 Female Rejected
126 Female Rejected
127 Female Rejected
128 Female Rejected
129 Female Rejected
130 Female Rejected
131 Female Rejected
132 Female Rejected
133 Female Rejected
134 Female Rejected
135 Female Rejected
136 Female Rejected
137 Female Rejected
138 Female Rejected
139 Female Rejected
140 Female Rejected
141 Female Rejected
142 Female Rejected
143 Female Rejected
144 Female Rejected
145 Female Rejected
146 Female Rejected
147 Female Rejected
148 Female Rejected
149 Female Rejected
150 Female Rejected
151 Female Rejected
152 Female Rejected
153 Female Rejected
154 Female Rejected
155 Female Rejected
156 Female Rejected
157 Female Rejected
158 Female Rejected
159 Female Rejected
160 Female Rejected
161 Female Rejected
162 Female Rejected
163 Female Rejected
164 Female Rejected
165 Female Rejected
166 Female Rejected
167 Female Rejected
168 Female Rejected
169 Female Rejected
170 Female Rejected
171 Female Rejected
172 Female Rejected
173 Female Rejected
174 Female Rejected
175 Female Rejected
176 Female Rejected
177 Female Rejected
178 Female Rejected
179 Female Rejected
180 Female Rejected
181 Female Rejected
182 Female Rejected
183 Female Rejected
184 Female Rejected
185 Female Rejected
186 Female Rejected
187 Female Rejected
188 Female Rejected
189 Female Rejected
190 Female Rejected
191 Female Rejected
192 Female Rejected
193 Female Rejected
194 Female Rejected
195 Female Rejected
196 Female Rejected
197 Female Rejected
198 Female Rejected
199 Female Rejected
200 Female Rejected
201 Female Rejected
202 Female Rejected
203 Female Rejected
204 Female Rejected
205 Female Rejected
206 Female Rejected
207 Female Rejected
208 Female Rejected
209 Female Rejected
210 Female Rejected
211 Female Rejected
212 Female Rejected
213 Female Rejected
214 Female Rejected
215 Female Rejected
216 Female Rejected
217 Female Rejected
218 Female Rejected
219 Female Rejected
220 Female Rejected
221 Female Rejected
222 Female Rejected
223 Female Rejected
224 Female Rejected
225 Female Rejected
226 Female Rejected
227 Female Rejected
228 Female Rejected
229 Female Rejected
230 Female Rejected
231 Female Rejected
232 Female Rejected
233 Female Rejected
234 Female Rejected
235 Female Rejected
236 Female Rejected
237 Female Rejected
238 Female Rejected
239 Female Rejected
240 Female Rejected
241 Female Rejected
242 Female Rejected
243 Female Rejected
244 Female Rejected
245 Female Rejected
246 Female Rejected
247 Female Rejected
248 Female Rejected
249 Female Rejected
250 Female Rejected
251 Female Rejected
252 Female Rejected
253 Female Rejected
254 Female Rejected
255 Female Rejected
256 Female Rejected
257 Female Rejected
258 Female Rejected
259 Female Rejected
260 Female Rejected
261 Female Rejected
262 Female Rejected
263 Female Rejected
264 Female Rejected
265 Female Rejected
266 Female Rejected
267 Female Rejected
268 Female Rejected
269 Female Rejected
270 Female Rejected
271 Female Rejected
272 Female Rejected
273 Female Rejected
274 Female Rejected
275 Female Rejected
276 Female Rejected
277 Female Rejected
278 Female Rejected
279 Female Rejected
280 Female Rejected
281 Female Rejected
282 Female Rejected
283 Female Rejected
284 Female Rejected
285 Female Rejected
286 Female Rejected
287 Female Rejected
288 Female Rejected
289 Female Rejected
290 Female Rejected
291 Female Rejected
292 Female Rejected
293 Female Rejected
294 Female Rejected
295 Female Rejected
296 Female Rejected
297 Female Rejected
298 Female Rejected
299 Female Rejected
300 Female Rejected
301 Female Rejected
302 Female Rejected
303 Female Rejected
304 Female Rejected
305 Female Rejected
306 Female Rejected
307 Female Rejected
308 Female Rejected
309 Female Rejected
310 Female Rejected
311 Female Rejected
312 Female Rejected
313 Female Rejected
314 Female Rejected
315 Female Rejected
316 Female Rejected
317 Female Rejected
318 Female Rejected
319 Female Rejected
320 Female Rejected
321 Female Rejected
322 Female Rejected
323 Female Rejected
324 Female Rejected
325 Female Rejected
326 Female Rejected
327 Female Rejected
328 Female Rejected
329 Female Rejected
330 Female Rejected
331 Female Rejected
332 Female Rejected
333 Female Rejected
334 Female Rejected
335 Female Rejected
336 Female Rejected
337 Female Rejected
338 Female Rejected
339 Female Rejected
340 Female Rejected
341 Female Rejected
342 Female Rejected
343 Female Rejected
344 Female Rejected
345 Female Rejected
346 Female Rejected
347 Female Rejected
348 Female Rejected
349 Female Rejected
350 Female Rejected
351 Female Rejected
352 Female Rejected
353 Female Rejected
354 Female Rejected
355 Female Rejected
356 Female Rejected
357 Female Rejected
358 Female Rejected
359 Female Rejected
360 Female Rejected
361 Female Rejected
362 Female Rejected
363 Female Rejected
364 Female Rejected
365 Female Rejected
366 Female Rejected
367 Female Rejected
368 Female Rejected
369 Female Rejected
370 Female Rejected
371 Female Rejected
372 Female Rejected
373 Female Rejected
374 Female Rejected
375 Female Rejected
376 Female Rejected
377 Female Rejected
378 Female Rejected
379 Female Rejected
380 Female Rejected
381 Female Rejected
382 Female Rejected
383 Female Rejected
384 Female Rejected
385 Female Rejected
386 Female Rejected
387 Female Rejected
388 Female Rejected
389 Female Rejected
390 Female Rejected
     Male Rejected
1     Male Rejected
2     Male Rejected
3     Male Rejected
4     Male Rejected
5     Male Rejected
6     Male Rejected
7     Male Rejected
8     Male Rejected
9     Male Rejected
10    Male Rejected
11    Male Rejected
12    Male Rejected
13    Male Rejected
14    Male Rejected
15    Male Rejected
16    Male Rejected
17    Male Rejected
18    Male Rejected
19    Male Rejected
20    Male Rejected
21    Male Rejected
22    Male Rejected
23    Male Rejected
24    Male Rejected
25    Male Rejected
26    Male Rejected
27    Male Rejected
28    Male Rejected
29    Male Rejected
30    Male Rejected
31    Male Rejected
32    Male Rejected
33    Male Rejected
34    Male Rejected
35    Male Rejected
36    Male Rejected
37    Male Rejected
38    Male Rejected
39    Male Rejected
40    Male Rejected
41    Male Rejected
42    Male Rejected
43    Male Rejected
44    Male Rejected
45    Male Rejected
46    Male Rejected
47    Male Rejected
48    Male Rejected
49    Male Rejected
50    Male Rejected
51    Male Rejected
52    Male Rejected
53    Male Rejected
54    Male Rejected
55    Male Rejected
56    Male Rejected
57    Male Rejected
58    Male Rejected
59    Male Rejected
60    Male Rejected
61    Male Rejected
62    Male Rejected
63    Male Rejected
64    Male Rejected
65    Male Rejected
66    Male Rejected
67    Male Rejected
68    Male Rejected
69    Male Rejected
70    Male Rejected
71    Male Rejected
72    Male Rejected
73    Male Rejected
74    Male Rejected
75    Male Rejected
76    Male Rejected
77    Male Rejected
78    Male Rejected
79    Male Rejected
80    Male Rejected
81    Male Rejected
82    Male Rejected
83    Male Rejected
84    Male Rejected
85    Male Rejected
86    Male Rejected
87    Male Rejected
88    Male Rejected
89    Male Rejected
90    Male Rejected
91    Male Rejected
92    Male Rejected
93    Male Rejected
94    Male Rejected
95    Male Rejected
96    Male Rejected
97    Male Rejected
98    Male Rejected
99    Male Rejected
100   Male Rejected
101   Male Rejected
102   Male Rejected
103   Male Rejected
104   Male Rejected
105   Male Rejected
106   Male Rejected
107   Male Rejected
108   Male Rejected
109   Male Rejected
110   Male Rejected
111   Male Rejected
112   Male Rejected
113   Male Rejected
114   Male Rejected
115   Male Rejected
116   Male Rejected
117   Male Rejected
118   Male Rejected
119   Male Rejected
120   Male Rejected
121   Male Rejected
122   Male Rejected
123   Male Rejected
124   Male Rejected
125   Male Rejected
126   Male Rejected
127   Male Rejected
128   Male Rejected
129   Male Rejected
130   Male Rejected
131   Male Rejected
132   Male Rejected
133   Male Rejected
134   Male Rejected
135   Male Rejected
136   Male Rejected
137   Male Rejected
138   Male Rejected
139   Male Rejected
140   Male Rejected
141   Male Rejected
142   Male Rejected
143   Male Rejected
144   Male Rejected
145   Male Rejected
146   Male Rejected
147   Male Rejected
148   Male Rejected
149   Male Rejected
150   Male Rejected
151   Male Rejected
152   Male Rejected
153   Male Rejected
154   Male Rejected
155   Male Rejected
156   Male Rejected
157   Male Rejected
158   Male Rejected
159   Male Rejected
160   Male Rejected
161   Male Rejected
162   Male Rejected
163   Male Rejected
164   Male Rejected
165   Male Rejected
166   Male Rejected
167   Male Rejected
168   Male Rejected
169   Male Rejected
170   Male Rejected
171   Male Rejected
172   Male Rejected
173   Male Rejected
174   Male Rejected
175   Male Rejected
176   Male Rejected
177   Male Rejected
178   Male Rejected
179   Male Rejected
180   Male Rejected
181   Male Rejected
182   Male Rejected
183   Male Rejected
184   Male Rejected
185   Male Rejected
186   Male Rejected
187   Male Rejected
188   Male Rejected
189   Male Rejected
190   Male Rejected
191   Male Rejected
192   Male Rejected
193   Male Rejected
194   Male Rejected
195   Male Rejected
196   Male Rejected
197   Male Rejected
198   Male Rejected
199   Male Rejected
200   Male Rejected
201   Male Rejected
202   Male Rejected
203   Male Rejected
204   Male Rejected
/ 59

Permutation test for independencedlong$GenderRandomized <- sample(dlong$Gender)
dlong

      Gender    Admit GenderRandomized
   Female Admitted             Male
1   Female Admitted           Female
2   Female Admitted           Female
3   Female Admitted             Male
4   Female Admitted           Female
5   Female Admitted           Female
6   Female Admitted             Male
7   Female Admitted           Female
8   Female Admitted           Female
9   Female Admitted             Male
10  Female Admitted             Male
11  Female Admitted             Male
12  Female Admitted           Female
13  Female Admitted           Female
14  Female Admitted             Male
15  Female Admitted           Female
16  Female Admitted           Female
17  Female Admitted             Male
18  Female Admitted             Male
19  Female Admitted           Female
20  Female Admitted           Female
21  Female Admitted             Male
22  Female Admitted             Male
23  Female Admitted             Male
24  Female Admitted           Female
25  Female Admitted             Male
26  Female Admitted           Female
27  Female Admitted             Male
28  Female Admitted           Female
29  Female Admitted           Female
30  Female Admitted           Female
31  Female Admitted             Male
32  Female Admitted           Female
33  Female Admitted           Female
34  Female Admitted             Male
35  Female Admitted           Female
36  Female Admitted           Female
37  Female Admitted           Female
38  Female Admitted             Male
39  Female Admitted             Male
40  Female Admitted             Male
41  Female Admitted           Female
42  Female Admitted             Male
43  Female Admitted           Female
44  Female Admitted           Female
45  Female Admitted             Male
46  Female Admitted           Female
47  Female Admitted           Female
48  Female Admitted           Female
49  Female Admitted           Female
50  Female Admitted           Female
51  Female Admitted             Male
52  Female Admitted           Female
53  Female Admitted           Female
54  Female Admitted           Female
55  Female Admitted           Female
56  Female Admitted             Male
57  Female Admitted           Female
58  Female Admitted           Female
59  Female Admitted             Male
60  Female Admitted           Female
61  Female Admitted           Female
62  Female Admitted           Female
63  Female Admitted           Female
64  Female Admitted           Female
65  Female Admitted           Female
66  Female Admitted             Male
67  Female Admitted           Female
68  Female Admitted           Female
69  Female Admitted           Female
70  Female Admitted           Female
71  Female Admitted           Female
72  Female Admitted             Male
73  Female Admitted           Female
74  Female Admitted             Male
75  Female Admitted           Female
76  Female Admitted             Male
77  Female Admitted           Female
78  Female Admitted           Female
79  Female Admitted           Female
80  Female Admitted           Female
81  Female Admitted           Female
82  Female Admitted             Male
83  Female Admitted           Female
84  Female Admitted           Female
85  Female Admitted             Male
86  Female Admitted           Female
87  Female Admitted             Male
88  Female Admitted           Female
89  Female Admitted           Female
90  Female Admitted           Female
91  Female Admitted           Female
92  Female Admitted           Female
93  Female Admitted             Male
94  Female Admitted           Female
95  Female Admitted           Female
96  Female Admitted           Female
97  Female Admitted             Male
98  Female Admitted           Female
99  Female Admitted           Female
100 Female Admitted           Female
101 Female Admitted           Female
102 Female Admitted             Male
103 Female Admitted           Female
104 Female Admitted           Female
105 Female Admitted             Male
106 Female Admitted           Female
107 Female Admitted             Male
108 Female Admitted             Male
109 Female Admitted             Male
110 Female Admitted           Female
111 Female Admitted           Female
112 Female Admitted           Female
113 Female Admitted             Male
114 Female Admitted           Female
115 Female Admitted           Female
116 Female Admitted           Female
117 Female Admitted             Male
118 Female Admitted           Female
119 Female Admitted           Female
120 Female Admitted           Female
121 Female Admitted             Male
122 Female Admitted           Female
123 Female Admitted           Female
124 Female Admitted           Female
125 Female Admitted           Female
126 Female Admitted             Male
127 Female Admitted             Male
128 Female Admitted             Male
129 Female Admitted             Male
130 Female Admitted           Female
131 Female Admitted             Male
132 Female Admitted           Female
133 Female Admitted           Female
134 Female Admitted           Female
135 Female Admitted             Male
136 Female Admitted             Male
137 Female Admitted             Male
138 Female Admitted             Male
139 Female Admitted           Female
140 Female Admitted           Female
141 Female Admitted           Female
142 Female Admitted           Female
143 Female Admitted             Male
144 Female Admitted             Male
145 Female Admitted           Female
146 Female Admitted           Female
147 Female Admitted             Male
148 Female Admitted           Female
149 Female Admitted           Female
150 Female Admitted             Male
151 Female Admitted           Female
152 Female Admitted           Female
153 Female Admitted           Female
154 Female Admitted           Female
155 Female Admitted           Female
156 Female Admitted             Male
157 Female Admitted             Male
158 Female Admitted           Female
159 Female Admitted             Male
160 Female Admitted           Female
161 Female Admitted           Female
162 Female Admitted             Male
163 Female Admitted           Female
164 Female Admitted           Female
165 Female Admitted           Female
166 Female Admitted           Female
167 Female Admitted           Female
168 Female Admitted             Male
169 Female Admitted           Female
170 Female Admitted             Male
171 Female Admitted             Male
172 Female Admitted             Male
173 Female Admitted           Female
174 Female Admitted           Female
175 Female Admitted           Female
176 Female Admitted             Male
177 Female Admitted           Female
178 Female Admitted           Female
179 Female Admitted             Male
180 Female Admitted           Female
181 Female Admitted             Male
182 Female Admitted           Female
183 Female Admitted             Male
184 Female Admitted           Female
185 Female Admitted             Male
186 Female Admitted           Female
187 Female Admitted           Female
188 Female Admitted           Female
189 Female Admitted           Female
190 Female Admitted           Female
191 Female Admitted           Female
192 Female Admitted           Female
193 Female Admitted           Female
194 Female Admitted           Female
195 Female Admitted             Male
196 Female Admitted             Male
197 Female Admitted           Female
198 Female Admitted           Female
199 Female Admitted           Female
200 Female Admitted           Female
201 Female Admitted           Female
     Male Admitted           Female
1     Male Admitted             Male
2     Male Admitted           Female
3     Male Admitted           Female
4     Male Admitted             Male
5     Male Admitted             Male
6     Male Admitted           Female
7     Male Admitted             Male
8     Male Admitted           Female
9     Male Admitted           Female
10    Male Admitted             Male
11    Male Admitted           Female
12    Male Admitted           Female
13    Male Admitted             Male
14    Male Admitted             Male
15    Male Admitted             Male
16    Male Admitted           Female
17    Male Admitted           Female
18    Male Admitted           Female
19    Male Admitted           Female
20    Male Admitted           Female
21    Male Admitted           Female
22    Male Admitted             Male
23    Male Admitted             Male
24    Male Admitted             Male
25    Male Admitted           Female
26    Male Admitted           Female
27    Male Admitted           Female
28    Male Admitted           Female
29    Male Admitted             Male
30    Male Admitted           Female
31    Male Admitted             Male
32    Male Admitted           Female
33    Male Admitted           Female
34    Male Admitted           Female
35    Male Admitted           Female
36    Male Admitted           Female
37    Male Admitted           Female
38    Male Admitted             Male
39    Male Admitted           Female
40    Male Admitted             Male
41    Male Admitted           Female
42    Male Admitted           Female
43    Male Admitted             Male
44    Male Admitted             Male
45    Male Admitted           Female
46    Male Admitted             Male
47    Male Admitted             Male
48    Male Admitted           Female
49    Male Admitted             Male
50    Male Admitted             Male
51    Male Admitted           Female
52    Male Admitted             Male
53    Male Admitted           Female
54    Male Admitted             Male
55    Male Admitted           Female
56    Male Admitted             Male
57    Male Admitted           Female
58    Male Admitted           Female
59    Male Admitted             Male
60    Male Admitted           Female
61    Male Admitted             Male
62    Male Admitted           Female
63    Male Admitted           Female
64    Male Admitted           Female
65    Male Admitted             Male
66    Male Admitted             Male
67    Male Admitted           Female
68    Male Admitted           Female
69    Male Admitted           Female
70    Male Admitted           Female
71    Male Admitted             Male
72    Male Admitted           Female
73    Male Admitted           Female
74    Male Admitted             Male
75    Male Admitted             Male
76    Male Admitted             Male
77    Male Admitted           Female
78    Male Admitted             Male
79    Male Admitted             Male
80    Male Admitted           Female
81    Male Admitted             Male
82    Male Admitted           Female
83    Male Admitted           Female
84    Male Admitted           Female
85    Male Admitted           Female
86    Male Admitted             Male
87    Male Admitted           Female
88    Male Admitted           Female
89    Male Admitted           Female
90    Male Admitted             Male
91    Male Admitted           Female
92    Male Admitted             Male
93    Male Admitted             Male
94    Male Admitted           Female
95    Male Admitted           Female
96    Male Admitted           Female
97    Male Admitted           Female
98    Male Admitted           Female
99    Male Admitted           Female
100   Male Admitted             Male
101   Male Admitted           Female
102   Male Admitted             Male
103   Male Admitted           Female
104   Male Admitted           Female
105   Male Admitted             Male
106   Male Admitted             Male
107   Male Admitted           Female
108   Male Admitted           Female
109   Male Admitted           Female
110   Male Admitted             Male
111   Male Admitted           Female
112   Male Admitted           Female
113   Male Admitted             Male
114   Male Admitted             Male
115   Male Admitted           Female
116   Male Admitted           Female
117   Male Admitted             Male
118   Male Admitted           Female
119   Male Admitted             Male
   Female Rejected             Male
1   Female Rejected           Female
2   Female Rejected             Male
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/ 59

Permutation test for independence

xtabs(~ GenderRandomized + Admit, dlong)

                Admit
GenderRandomized Admitted Rejected
          Female      205      388
          Male        117      208

log(oddsratio(xtabs(~ GenderRandomized + Admit, dlong)))

[1] -0.06263122

49 / 59

Permutation test for independence

We can repeat this process as many times as we want
These are simulations from the distribution of log odds ratio (under independence)

sim_logOR <-
    replicate(10000,
              {
                  dlong$GenderRandomized <- sample(dlong$Gender)
                  log(oddsratio(xtabs(~ GenderRandomized + Admit, dlong)))
              })

str(sim_logOR)

 num [1:10000] 0.3191 -0.187 -0.1249 0.1474 -0.0418 ...

50 / 59

Permutation test for independence

histogram(~ sim_logOR, nint = 30)

plot of chunk unnamed-chunk-43

51 / 59

Permutation test for independence

histogram(~ sim_logOR, nint = 30) + layer(panel.abline(v = log(oddsratio(X)), col = "red"))

plot of chunk unnamed-chunk-44

52 / 59

Permutation test for independence

ecdfplot(~ sim_logOR, grid = TRUE) + layer(panel.abline(v = log(oddsratio(X)), col = "red"))

plot of chunk unnamed-chunk-45

53 / 59

Permutation test for independenceIs the observed value unusual?
54 / 59

Permutation test for independence

Is the observed value unusual?
Roughly 20% of simulations had log OR less than the observed value
So we should probably not consider this very strong evidence of gender bias

54 / 59

Permutation test for independence

Is the observed value unusual?
Roughly 20% of simulations had log OR less than the observed value
So we should probably not consider this very strong evidence of gender bias
This procedure is known as Fisher's Exact Test of Independence

54 / 59

Permutation test for independence

Department E had the most negative log odds ratio

X <- t(UCBAdmissions[, c(2, 1), "E"])
X

        Admit
Gender   Admitted Rejected
  Female       94      299
  Male         53      138

log(oddsratio(X))

[1] -0.200187

55 / 59

Permutation test for independence

d <- array2DF(X)
dlong <- d[rep(1:4, d$Value) , c("Gender", "Admit")]
sim_logOR <-
    replicate(10000,
              {
                  dlong$GenderRandomized <- sample(dlong$Gender)
                  log(oddsratio(xtabs(~ GenderRandomized + Admit, dlong)))
              })

56 / 59

Permutation test for independence

ecdfplot(~ sim_logOR, grid = TRUE) + layer(panel.abline(v = log(oddsratio(X)), col = "red"))

plot of chunk unnamed-chunk-48

57 / 59

ExerciseTry this for department A
58 / 59

Exercise

Try this for department A
Do we really need to do this permutation?
Note that if row totals are fixed, log OR depends only on $X_{11}$

58 / 59

Exercise

Try this for department A
Do we really need to do this permutation?
Note that if row totals are fixed, log OR depends only on $X_{11}$
Exercise: Explicitly calculate distribution of $X_{11}$ given row and column totals

58 / 59

Questions?59 / 59

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Categorical Data

Statistics I — Data Exploration

Deepayan Sarkar

Motivation

Example: Categorical X

Example: Categorical X

Example: Categorical X

Example: Numeric X

Example: Numeric X

Categorical response

Categorical response

Contingency tables summarizing categorical data

Example: Race and Gender

Example: Race and Gender

Example: Race and Gender

Example: Race and Gender

Representing contingency tables

Representing contingency tables

Example: UCBAdmissions

Example: UCBAdmissions

Example: UCBAdmissions

Example: UCBAdmissions

Example: UCBAdmissions

Example: UCBAdmissions

Example: UCBAdmissions

Example: UCBAdmissions

Example: UCBAdmissions

Example: UCBAdmissions

Summarizing conditional distribution of Y

Summarizing conditional distribution of Y

Summarizing conditional distribution of Y

Odds

Odds

Odds

Odds as a transformation

Odds as a transformation

Log odds

Log odds

Example: UN data

Example: UN data

Odds ratio

Odds ratio

Odds ratio

Odds ratio

Example: UCBAdmissions revisited

Example: UCBAdmissions revisited

Example: UCBAdmissions revisited

Example: Hypertension vs Age

Example: Hypertension vs Age

Example: Hypertension vs Age

Example: Hypertension vs Age

Example: Hypertension vs Age

Example: Titanic data

Example: Titanic data

Example: Titanic data

Are two attributes independent?

Are two attributes independent?

Are two attributes independent?

Are two attributes independent?

Are two attributes independent?

Are two attributes independent?

Are two attributes independent?

Are two attributes independent?

Are two attributes independent?

Are two attributes independent?

Are two attributes independent?

Are two attributes independent?

Are two attributes independent?

Permutation test for independence

Permutation test for independence

Permutation test for independence

Permutation test for independence

Permutation test for independence

Permutation test for independence

Permutation test for independence

Permutation test for independence

Permutation test for independence

Permutation test for independence

Permutation test for independence

Permutation test for independence

Example: Categorical $X$

Example: Categorical $X$

Example: Categorical $X$

Example: Numeric $X$

Example: Numeric $X$

Example: `UCBAdmissions`

Example: `UCBAdmissions`

Example: `UCBAdmissions`

Example: `UCBAdmissions`

Example: `UCBAdmissions`

Example: `UCBAdmissions`

Example: `UCBAdmissions`

Example: `UCBAdmissions`

Example: `UCBAdmissions`

Example: `UCBAdmissions`

Summarizing conditional distribution of $Y$

Summarizing conditional distribution of $Y$

Summarizing conditional distribution of $Y$

Example: `UCBAdmissions` revisited

Example: `UCBAdmissions` revisited

Example: `UCBAdmissions` revisited

Example: `Titanic` data

Example: `Titanic` data

Example: `Titanic` data