Unit 11 - Inferences for Distributions of Categorical Data

chi-square goodness-of-fit test (+hypotheses)

allows us to determine whether a hypothesized distribution seems valid

H₀: the stated distribution of (the categorical variable in the population of interest) is correct
H_a: the stated distribution of (the categorical variable in the population of interest) is not correct ^{_{(always two-sided!) example in pic
-compare observed counts from our sample to the expected counts (this is for all x² tests)}}

chi-square statistic

measures how far apart the observed and expected counts are _{^{(for GOF and homogeneity)}}

★when showing calculations, write out the work for _{^{(at least)}} 2 categories
^get contribution of cell by doing the individual (obs-exp)²/exp for that cell

Drawing chi-square curve

• right skewed _{^{(only + values, x² can never be -)}}

• label 0 to wtv _{^{(I recommend twice the x² value)}}

• label tick at x², shade right

• write x²=# below the curve

• label curve with x²_df (df for GOF is # of categories - 1) ^{_{(# of categories of the categorical variable)}}

incr df, shape less skewed (still skewed tho)

How to get p-value for chi-square test (GOF, homogeneity, or independence)

do x²cdf _{^{(from [2nd] [vars] on calc)}}
put in lower x² value, upper 1000, and df
p-value always area under the curve to the right of x²
write probability statement → ex) P(x²>#)

Conditions for chi-squared goodness-of-fit test or chi-square test for homogeneity or chi-square test for independence

• random sampling/assignment (homogeneity - do for each population!!)

• 10% (n≤0.1N) _{^{(independent observations)}} (homogeneity - do for each population!!) (skip if experiment)

• Large counts (all expected counts ≥5)

+categorical data!! _{^{/ expected count from proportion * sample size. LEAVE THE DECIMAL!}}

!!! (flip for chi-square info cheat sheet/summary)
do homo/indep tests on same data -> same x² and p-value! (diff hypotheses, conditions, & conclusion tho)

x² distribution cannot be negative (always +, curve drawn w ticks above 0)
x² is unitless

don’t need to define anything in chi-square tests b/c hypotheses are in words

calculator: [stat] [Tests] x²GOF-Test (put data into Lists, say which is Observed/Expected, put df)
^do x²-Test for homogeneity and inference (put data into matrix ([2nd] [x^-1] then edit. don’t include totals. to change which matrix you put in for the test, press [2nd] [x^-1] then enter on the matrix you want to fill observed/expected)

multiple comparisons - hard to do many comparisons at once
^inference for 2-way tables (need counts!)_{^{→do overall test to see if there is good evidence of any differences among the parameters we want to compare, then detailed follow-up analysis to decide which of the parameters differ and to estimate how large the differences are (good evidence of difference(s)? what are the differences (details)?)}}
^compare observed counts from 2-way table with expected counts

Hypotheses for chi-square test for homogeneity

H₀: there is no difference in the distribution of [categorical variable] for [the several populations or treatments]

H_a: there is a difference ^

_{^{^ex: sports goals for categorical var; females and males at the university for several populations}}

How to do conditional distribution for two-way table (homogeneity/independence)

do each cell out of the total. total must be from a population (think of it as finding proportions by breaking down the populations into different parts of a categorical variable) ← use segmented bar chart or side-by-side bar chart

How to find expected counts when H₀ is true
(homogeneity/independence tests)

How to get df for chi-squared test for homogeneity/independence

df = (# of rows - 1)(# of columns - 1)

Hypotheses for chi-square test for independence

H₀: there is no association btwn (2 categorical variables) in (the population of interest)

H_a: there is an association ^

_{^{*what is the relationship of the variables?}}

Interpret p-value for homogeneity and independence

Assuming (null is true), there is (p-value) probability of…

homogeneity: …observing a difference in (categorical variable) as large or larger than the difference observed in the study by chance alone

independence: …getting a random sample of (n population) with an association as strong or stronger than the one found in the study

Difference between the chi-square tests

GOF: 1 variable, 1 pop

homogeneity: 1 var, multiple populations/samples/treatments

independence: multiple vars, 1 population/sample

!!! draw bar chart → can do percentage for y-axis, do out of POPULATION(S)

say categorical variable(s) and population(s) in hypotheses!

follow-up analysis → what is contributing the most to the chi-square statistic? (get from doing the individual calculation for that cell from chi-square. (observed-expected)²/expected for just that cell) (cell highest contribution -> deviates most from null) (when interpret, say the # for the contribution & compare the observed and expected value for the cell that contributes most. this cell suggests (H_a) is true)

experiment can be homo/indep. homo if only 1 variable but several treatments. indep if finding association btwn two variables.

random assignment → don’t need to do 10% (purpose of random assignment: create roughly equivalent groups at the start of the study + reduce confounding variables)
^different assignments into groups bc experiment -> multiple treatments-> homogeneity

show work to get expected counts (esp for GOF - percentage times the sample size)