Let’s load dataset survey from package
MASS.
library(MASS)
data(survey)
survey## Sex Wr.Hnd NW.Hnd W.Hnd Fold Pulse Clap Exer Smoke Height M.I Age
## 1 Female 18.5 18.0 Right R on L 92 Left Some Never 173.00 Metric 18.250
## 2 Male 19.5 20.5 Left R on L 104 Left None Regul 177.80 Imperial 17.583
## 3 Male 18.0 13.3 Right L on R 87 Neither None Occas NA <NA> 16.917
## 4 Male 18.8 18.9 Right R on L NA Neither None Never 160.00 Metric 20.333
## 5 Male 20.0 20.0 Right Neither 35 Right Some Never 165.00 Metric 23.667
## 6 Female 18.0 17.7 Right L on R 64 Right Some Never 172.72 Imperial 21.000
## 7 Male 17.7 17.7 Right L on R 83 Right Freq Never 182.88 Imperial 18.833
## 8 Female 17.0 17.3 Right R on L 74 Right Freq Never 157.00 Metric 35.833
## 9 Male 20.0 19.5 Right R on L 72 Right Some Never 175.00 Metric 19.000
## 10 Male 18.5 18.5 Right R on L 90 Right Some Never 167.00 Metric 22.333
## 11 Female 17.0 17.2 Right L on R 80 Right Freq Never 156.20 Imperial 28.500
## 12 Male 21.0 21.0 Right R on L 68 Left Freq Never NA <NA> 18.250
## 13 Female 16.0 16.0 Right L on R NA Right Some Never 155.00 Metric 18.750
## 14 Female 19.5 20.2 Right L on R 66 Neither Some Never 155.00 Metric 17.500
## 15 Male 16.0 15.5 Right R on L 60 Right Some Never NA <NA> 17.167
## 16 Female 17.5 17.0 Right R on L NA Right Freq Never 156.00 Metric 17.167
## 17 Female 18.0 18.0 Right L on R 89 Neither Freq Never 157.00 Metric 19.333
## 18 Male 19.4 19.2 Left R on L 74 Right Some Never 182.88 Imperial 18.333
## 19 Male 20.5 20.5 Right L on R NA Left Some Never 190.50 Imperial 19.750
## 20 Male 21.0 20.9 Right R on L 78 Right Freq Never 177.00 Metric 17.917
## 21 Male 21.5 22.0 Right R on L 72 Left Freq Never 190.50 Imperial 17.917
## 22 Male 20.1 20.7 Right L on R 72 Right Freq Never 180.34 Imperial 18.167
## 23 Male 18.5 18.0 Right L on R 64 Right Freq Never 180.34 Imperial 17.833
## 24 Male 21.5 21.2 Right R on L 62 Right Some Never 184.00 Metric 18.250
## 25 Female 17.0 17.5 Right R on L 64 Left Some Never NA <NA> 19.167
## 26 Male 18.5 18.5 Right Neither 90 Neither Some Never NA <NA> 17.583
## 27 Male 21.0 20.7 Right R on L 90 Right Some Never 172.72 Imperial 17.500
## 28 Male 20.8 21.4 Right R on L 62 Neither Freq Never 175.26 Imperial 18.083
## 29 Male 17.8 17.8 Right L on R 76 Neither Freq Never NA <NA> 21.917
## 30 Male 19.5 19.5 Right L on R 79 Right Some Never 167.00 Metric 19.250
## 31 Female 18.5 18.0 Right R on L 76 Right None Occas NA <NA> 41.583
## 32 Male 18.8 18.2 Right L on R 78 Right Freq Never 180.00 Metric 17.500
## 33 Female 17.1 17.5 Right R on L 72 Right Freq Heavy 166.40 Imperial 39.750
## 34 Male 20.1 20.0 Right R on L 70 Right Some Never 180.00 Metric 17.167
## 35 Male 18.0 19.0 Right L on R 54 Neither Some Regul NA <NA> 17.750
## 36 Male 22.2 21.0 Right L on R 66 Right Freq Occas 190.00 Metric 18.000
## 37 Female 16.0 16.5 Right L on R NA Right Some Never 168.00 Metric 19.000
## 38 Male 19.4 18.5 Right R on L 72 Neither Freq Never 182.50 Metric 17.917
## 39 Male 22.0 22.0 Right R on L 80 Right Some Never 185.00 Metric 35.500
## 40 Male 19.0 19.0 Right R on L NA Neither Freq Occas 171.00 Metric 19.917
## 41 Female 17.5 16.0 Right L on R NA Right Some Never 169.00 Metric 17.500
## 42 Female 17.8 18.0 Right R on L 72 Right Some Never 154.94 Imperial 17.083
## 43 Male NA NA Right R on L 60 <NA> Some Never 172.00 Metric 28.583
## 44 Female 20.1 20.2 Right L on R 80 Right Some Never 176.50 Imperial 17.500
## 45 Female 13.0 13.0 <NA> L on R 70 Left Freq Never 180.34 Imperial 17.417
## 46 Male 17.0 17.5 Right R on L NA Neither Freq Never 180.34 Imperial 18.500
## 47 Male 23.2 22.7 Right L on R 84 Left Freq Regul 180.00 Metric 18.917
## 48 Male 22.5 23.0 Right R on L 96 Right None Never 170.00 Metric 19.417
## 49 Female 18.0 17.6 Right R on L 60 Right Some Occas 168.00 Metric 18.417
## 50 Female 18.0 17.9 Right R on L 50 Left None Never 165.00 Metric 30.750
## 51 Male 22.0 21.5 Left R on L 55 Left Freq Never 200.00 Metric 18.500
## 52 Male 20.5 20.0 Right L on R 68 Right Freq Never 190.00 Metric 17.500
## 53 Male 17.0 18.0 Right L on R 78 Left Some Never 170.18 Imperial 18.333
## 54 Male 20.5 19.5 Right L on R 56 Right Freq Never 179.00 Metric 17.417
## 55 Male 22.5 22.5 Right R on L 65 Right Freq Regul 182.00 Metric 20.000
## 56 Male 18.5 18.5 Right L on R NA Neither Freq Never 171.00 Metric 18.333
## 57 Female 15.5 15.4 Right R on L 70 Neither None Never 157.48 Imperial 17.167
## 58 Male 19.5 19.7 Right R on L 72 Right Freq Never NA <NA> 17.417
## 59 Male 19.5 19.0 Right L on R 62 Right Freq Never 177.80 Imperial 17.667
## 60 Male 20.6 21.0 Left L on R NA Left Freq Occas 175.26 Imperial 18.417
## 61 Male 22.8 23.2 Right R on L 66 Neither Freq Never 187.00 Metric 20.333
## 62 Female 18.5 18.2 Right R on L 72 Neither Freq Never 167.64 Imperial 17.333
## 63 Female 19.6 19.7 Right L on R 70 Right Freq Never 178.00 Metric 17.500
## 64 Female 18.7 18.0 Left L on R NA Left None Never 170.00 Metric 19.833
## 65 Female 17.3 18.0 Right L on R 64 Neither Freq Never 164.00 Metric 18.583
## 66 Male 19.5 19.8 Right Neither NA Right Freq Never 183.00 Metric 18.000
## 67 Female 19.0 19.1 Right L on R NA Neither Freq Never 172.00 Metric 30.667
## 68 Female 18.5 18.0 Right R on L 64 Right Freq Never NA <NA> 16.917
## 69 Male 19.0 19.0 Right L on R NA Right Some Never 180.00 Metric 19.917
## 70 Male 21.0 19.5 Right L on R 80 Left None <NA> NA <NA> 18.333
## 71 Female 18.0 17.5 Right L on R 64 Left Freq Never 170.00 Metric 17.583
## 72 Male 19.4 19.5 Right R on L NA Right Freq Heavy 176.00 Metric 17.833
## 73 Female 17.0 16.6 Right R on L 68 Right Some Never 171.00 Metric 17.667
## 74 Female 16.5 17.0 Right L on R 40 Left Freq Never 167.64 Imperial 17.417
## 75 Female 15.6 15.8 Right R on L 88 Left Some Never 165.00 Metric 17.750
## 76 Female 17.5 17.5 Right Neither 68 Right Freq Heavy 170.00 Metric 20.667
## 77 Female 17.0 17.6 Right L on R 76 Right Some Never 165.00 Metric 23.583
## 78 Female 18.6 18.0 Right L on R NA Neither Freq Heavy 165.10 Imperial 17.167
## 79 Female 18.3 18.5 Right R on L 68 Neither Some Never 165.10 Imperial 17.083
## 80 Male 20.0 20.5 Right L on R NA Right Freq Never 185.42 Imperial 18.750
## 81 Male 19.5 19.5 Left R on L 66 Left Some Never NA <NA> 16.750
## 82 Male 19.2 18.9 Right R on L 76 Right Freq Never 176.50 Imperial 20.167
## 83 Female 17.5 17.5 Right R on L 98 Left Freq Never NA <NA> 17.667
## [ reached 'max' / getOption("max.print") -- omitted 154 rows ]To performa a goodness-of-fit test, we want to compare whether an
observed frequency is similar to expected frequency or not. For example,
we have a hypothesis that 20% of people frequently exercise, 50% do some
exercise, and 30% do not exercise. We will use Exer
variable to test this hypothesis.
We will use function chisq.test(x, p), where
x is a frequency table, and p is a vector of
expected probability for each category.
ex_tab <- table(survey$Exer) # create a table to be analyzed
ex_tab # note the order of the categories.##
## Freq None Some
## 115 24 98chi <- chisq.test(ex_tab, p = c(20/100, 30/100, 50/100)) # We put exptected p for each category here. 0.2 or 20% for frequent exercise, 0.3 or 30% for no exercise, 0.5 or 50% for some exercise.
chi # value of the chi-square test##
## Chi-squared test for given probabilities
##
## data: ex_tab
## X-squared = 131.16, df = 2, p-value < 2.2e-16chi$observed # observed value (same as the input table)##
## Freq None Some
## 115 24 98chi$expected # expected value## Freq None Some
## 47.4 71.1 118.5The \(\chi^2\) value was significant, suggesting that the observed frequencies were not distributed as expected. Looking at the results, we could see that there were more people who exercise frequently (115) more than our expectation (20% or 47.6), and there were fewer the people who do no exercise (24) than what we expected (30% or 71.1).
Next we would like to consider whether exercise frequencies depends
on (or associated with) biological sex (Sex). The idea
behind the test is that if Sex has nothing to do with
exercise frequencies, then the distribution of exercise should be the
same in male and female (proportion, not expected value).
se_tab <- table(survey$Sex, survey$Exer)
se_tab##
## Freq None Some
## Female 49 11 58
## Male 65 13 40chi2 <- chisq.test(se_tab)
chi2##
## Pearson's Chi-squared test
##
## data: se_tab
## X-squared = 5.7184, df = 2, p-value = 0.05731chi2$observed##
## Freq None Some
## Female 49 11 58
## Male 65 13 40chi2$expected # we have the same expected values for both sexes because we have equal number of male and female. ##
## Freq None Some
## Female 57 12 49
## Male 57 12 49The test was not significant, but very close. When we compare the expected values to the observed values, we could see that more men seems to exercise frequently than women. On ther other hand, for people with some exercise, there seems to be more female than male. Although we cannot claim to find this pattern because the \(\chi^2\) was not significant, this pattern might worth looking into in a future study.
jmvIn jamovi’s package, we can use propTestN(), proportion
test (N outcomes), for a goodness-of-fit test.
library(jmv)
propTestN(data = survey,
var = Exer, # variable to be analyzed
expected = TRUE, #show expected values
ratio = c(20/100, 30/100, 50/100)
)##
## PROPORTION TEST (N OUTCOMES)
##
## Proportions - Exer
## ────────────────────────────────────────────────
## Level Count Proportion
## ────────────────────────────────────────────────
## Freq Observed 115 0.4852321
## Expected 47.40000 0.2000000
##
## None Observed 24 0.1012658
## Expected 71.10000 0.3000000
##
## Some Observed 98 0.4135021
## Expected 118.50000 0.5000000
## ────────────────────────────────────────────────
##
##
## χ² Goodness of Fit
## ────────────────────────────────
## χ² df p
## ────────────────────────────────
## 131.1561 2 < .0000001
## ────────────────────────────────For contingency table, we use contTables().
contTables(data = survey,
rows = Sex,
cols = Exer,
exp = TRUE) # show expected values##
## CONTINGENCY TABLES
##
## Contingency Tables
## ───────────────────────────────────────────────────────────────────────
## Sex Freq None Some Total
## ───────────────────────────────────────────────────────────────────────
## Female Observed 49 11 58 118
## Expected 57.00000 12.00000 49.00000 118.0000
##
## Male Observed 65 13 40 118
## Expected 57.00000 12.00000 49.00000 118.0000
##
## Total Observed 114 24 98 236
## Expected 114.00000 24.00000 98.00000 236.0000
## ───────────────────────────────────────────────────────────────────────
##
##
## χ² Tests
## ─────────────────────────────────────
## Value df p
## ─────────────────────────────────────
## χ² 5.718403 2 0.0573145
## N 236
## ─────────────────────────────────────Copyright © 2022 Kris Ariyabuddhiphongs