Background
The table() function
In reality, you will need to convert your data to fit the various nominal test
specifications. To do so, you may wish to enlist the help of the table()
function of base R which converts nominal records into count data.
Samples <- c("A", "B")
set.seed(42)
counts <- sample(Samples, size = 1000, replace = TRUE)
table(counts)
## counts
## A B
## 499 501
Aarhus University Biostatistics - Why? What? How? 5 / 19
Analyses Binomial Test
Purpose And Assumptions
Binomial Test
binom.test() in base R
Purpose:
To test whether the observed distribution of data values of a
binomial variable differ from what was expected.
H
0
The observed binomial data proportions do not differ
significantly from the expected proportions.
Assumptions:
Variable values are binomial.
The population is significantly larger than the sample.
The sample accurately represents the population.
Sampled values are independent (one value does not
influence another).
Aarhus University Biostatistics - Why? What? How? 7 / 19
Analyses Binomial Test
Minimal Working Example
We feed the binom.test() function a 1/4 (c(200, 800)) data set whilst
expecting the distribution to be 1 to 1 (p = 0.5).
binom.test(c(200, 800), p = 0.5)
##
## Exact binomial test
##
## data: c(200, 800)
## number of successes = 200, number of trials = 1000,
## p-value <2e-16
## alternative hypothesis: true probability of success is not equal to 0.5
## 95 percent confidence interval:
## 0.18 0.23
## sample estimates:
## probability of success
## 0.2
The result is significant (p ≈ 0).
Aarhus University Biostatistics - Why? What? How? 8 / 19
Analyses McNemar
Purpose And Assumptions
McNemar
mcnemar.test() in base R
Purpose:
To test whether there is a change in proportion of paired data.
H
0
The observed binomial data proportions do not differ
significantly between treatments/paired sets.
Assumptions:
Variable values are binomial.
The population is significantly larger than the sample.
The sample accurately represents the population.
Aarhus University Biostatistics - Why? What? How? 9 / 19
Analyses McNemar
Minimal Working Example
We feed the mcnemar.test() function a 1 to 1 (c(500, 500)) as well as a
a 1 to 4 (c(200, 800)) data set for the paired data sets.
Performance <- matrix(c(500, 500, 200, 800), nrow = 2)
Performance
## [,1] [,2]
## [1,] 500 200
## [2,] 500 800
mcnemar.test(Performance)
##
## McNemar's Chi-squared test with continuity
## correction
##
## data: Performance
## McNemar's chi-squared = 128, df = 1, p-value <2e-16
With a p-value of ≈ 0 the test concludes significantly.
Aarhus University Biostatistics - Why? What? How? 10 / 19
Analyses Cochran’s Q
Purpose And Assumptions
Cochran’s Q
cochrans.q() in the nonpar package
Purpose:
To test whether there are differences in matched sets of three
or more frequencies or proportions.
H
0
The observed proportions of data values in treatments is equal
among the matched sets.
Assumptions:
The first (dependent/response) variable is binomial.
The second (independent/predictor) variable is
nominal/categorical with three values.
The population is significantly larger than the sample.
The sample accurately represents the population.
Aarhus University Biostatistics - Why? What? How? 11 / 19
Analyses Cochran’s Q
Minimal Working Example
We feed the cochrans.q() function a 6 by 4 matrix of binomial values. The
first column represents our dependent, binomial variable. The remaining
columns represent our independent variable on three levels expressed as
binomial values.
CochranMatrix <- matrix(c(1, 1, 1, 1, 1, 1, 1, 1, 0,
1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1), 6,
4)
cochrans.q(CochranMatrix)
##
## Cochran's Q Test
##
## H0: There is no difference in the effectiveness of treatments.
## HA: There is a difference in the effectiveness of treatments.
##
## Q = 9.31578947368421
##
## Degrees of Freedom = 3
##
## Significance Level = 0.05
## The p-value is 0.0253739987887868
## There is enough evidence to conclude that the effectiveness of at least two treatments differ.
##
Aarhus University Biostatistics - Why? What? How? 12 / 19
Analyses Chi-Squared
Purpose And Assumptions
Chi-Squared
chisq.test() in base R
Purpose:
To tests whether distributions of categorical variables differ from
one another thus identifying whether they are related.
H
0
The distributions of nominal variables are equal.
Assumptions:
Variable values are nominal/categorical.
The population is significantly larger than the sample.
The sample accurately represents the population.
Sampled values are independent (one value does not
influence another).
Aarhus University Biostatistics - Why? What? How? 13 / 19
Analyses Chi-Squared
Minimal Working Example - One Sample Situation I
We feed the
chisq.test()
function an unbiased nominal set of three levels:
set.seed(42)
ChiMat1 <- table(sample(c("A", "B", "C"), 1000, replace = TRUE))
ChiMat1
##
## A B C
## 329 360 311
chisq.test(ChiMat1)
##
## Chi-squared test for given probabilities
##
## data: ChiMat1
## X-squared = 4, df = 2, p-value = 0.2
Obviously, the observed distribution does not differ from our expectation of
equally distributed proportions and so the test concludes non-significantly.
Aarhus University Biostatistics - Why? What? How? 14 / 19
Analyses Chi-Squared
Minimal Working Example - One Sample Situation II
We feed the chisq.test() function a skewed (towards "A") nominal set of
three levels:
set.seed(42)
ChiMat2 <- table(sample(c("A", "B", "C"), 1000, replace = TRUE,
prob = c(0.8, 0.1, 0.1)))
ChiMat2
##
## A B C
## 804 111 85
chisq.test(ChiMat2)
##
## Chi-squared test for given probabilities
##
## data: ChiMat2
## X-squared = 998, df = 2, p-value <2e-16
Obviously, the observed distribution does differ from our expectation of equally
distributed proportions and so the test concludes significantly.
Aarhus University Biostatistics - Why? What? How? 15 / 19
Analyses Chi-Squared
Minimal Working Example - Two Sample Situation
We feed the chisq.test() our unbiased as well as our skewed (towards
"A") nominal set of three levels to see whether their distributions differ
significantly.
ChiMatrix <- cbind(ChiMat1, ChiMat2)
ChiMatrix
## ChiMat1 ChiMat2
## A 329 804
## B 360 111
## C 311 85
chisq.test(ChiMatrix)
##
## Pearson's Chi-squared test
##
## data: ChiMatrix
## X-squared = 460, df = 2, p-value <2e-16
Clearly, they do differ significantly.
Aarhus University Biostatistics - Why? What? How? 16 / 19
Our Data Choice Of Variables
Variables We Can Use
Which variables in our Passer domesticus data set are nominal?
Site Index
Climate
Population Status
Colour
Sex
Nesting Site
Flock
Home Range
Predator Presence
Predator Type
All of these are nominal but some are binomial.
Aarhus University Biostatistics - Why? What? How? 18 / 19
Our Data Research Questions
Research Questions And Hypotheses
So which of our major research questions (seminar 6) can we answer?
Binomial Test
Sexual Dimorphism: Are the sexes
represented in equal proportions?
Predation: Are our sites dominated by
predators or not?
McNemar
Sexual Dimorphism: Compare sex
ratio over time (we need to generate
some new data for this using the
sample function).
Cochran’s Q
Sexual Dimorphism: Are colours
related to sex?
Predation: Are colours related to
predators?
Chi-Squared
Sexual Dimorphism: Are colours
related to sex?
Predation: Are colours or nesting sites
related to predators?
Aarhus University Biostatistics - Why? What? How? 19 / 19
