Chapter 2 Week 4

2.1 epi.2by2

  • Summary measures for count data presented in a 2 by 2 table
library(epiR)
epi.2by2(dat, 
         method = c("cohort.count", "cohort.time", "case.control", "cross.sectional"), 
         conf.level = 0.95, 
         units = 100, 
         outcome = c("as.columns","as.rows"))

  • dat: an object of class table containing the individual cell frequencies.

  • method: a character string indicating the study design on which the tabular data has been based.

    • Based on the study design specified by the user, appropriate measures of association, measures of effect in the exposed and measures of effect in the population are returned by the function.
cohort.count case.control
Measures of association
RR \(\checkmark\)
OR \(\checkmark\) \(\checkmark\)
Measures of effect in the exposed
ARisk \(\checkmark\) \(\checkmark\)
AFRisk \(\checkmark\) AFest
Measures of effect in the population
PARisk \(\checkmark\) \(\checkmark\)
PAFRisk \(\checkmark\) PAFest
chi-squared test
chisq.strata \(\checkmark\) \(\checkmark\)
chisq.crude \(\checkmark\) \(\checkmark\)
chisq.mh \(\checkmark\) \(\checkmark\)
Mantel-Haenszel (Woolf) test of homogeneity
RR.homog \(\checkmark\)
OR.homog \(\checkmark\) \(\checkmark\)

  • conf.level: magnitude of the returned confidence intervals. The default value is 0.95.

  • units: multiplier for prevalence and incidence (risk or rate) estimates. Default value is 100.

    • unit=1 Outcomes per population unit

    • unit=100 Outcomes per 100 population unit

    • The multiplier is applied to Attributable risk, Attributable risk in population, etc.

  • outcome: how the outcome variable is represented in the contingency table.

2.1.1 Example

# Example in A01
birthwt <- data.frame("Low" = c(21054, 27126), "Normal" = c(14442, 3804294), row.names = c("Dead at Year 1", "Alive at Year 1"))
birthwt
##                   Low  Normal
## Dead at Year 1  21054   14442
## Alive at Year 1 27126 3804294
epi.2by2(birthwt, 
         method = "cohort.count", 
         conf.level = 0.95, 
         units = 100, 
         outcome = "as.rows") # the outcome is represented as rows
##              Exposed +    Exposed -      Total
## Outcome +        21054        14442      35496
## Outcome -        27126      3804294    3831420
## Total            48180      3818736    3866916
## 
## Point estimates and 95% CIs:
## -------------------------------------------------------------------
## Inc risk ratio                               115.55 (113.35, 117.78)
## Odds ratio                                   204.45 (199.54, 209.49)
## Attrib risk *                                43.32 (42.88, 43.76)
## Attrib risk in population *                  0.54 (0.53, 0.55)
## Attrib fraction in exposed (%)               99.13 (99.12, 99.15)
## Attrib fraction in population (%)            58.80 (58.28, 59.31)
## -------------------------------------------------------------------
##  Test that OR = 1: chi2(1) = 981742.864 Pr>chi2 = <0.001
##  Wald confidence limits
##  CI: confidence interval
##  * Outcomes per 100 population units

2.2 chisq.test

  • performs chi-squared contingency table tests and goodness-of-fit tests.
chisq.test(x, y = NULL, correct = TRUE,
           p = rep(1/length(x), length(x)), rescale.p = FALSE,
           simulate.p.value = FALSE, B = 2000)

  • x, y = NULL: the input can be two numeric vectors as x and y (can both be factors), or a matrix as x.

  • correct: a logical indicating whether to apply continuity correction.

2.2.1 output components

chi <- chisq.test(birthwt)
chi
## 
##  Pearson's Chi-squared test with Yates' continuity correction
## 
## data:  birthwt
## X-squared = 981695, df = 1, p-value < 2.2e-16
# use dollar sign($) to specify output
chi$statistic # the value the chi-squared test statistic
## X-squared 
##  981695.2
chi$parameter # the degrees of freedom
## df 
##  1
chi$p.value
## [1] 0
chi$method
## [1] "Pearson's Chi-squared test with Yates' continuity correction"
chi$data.name
## [1] "birthwt"
chi$observed
##                   Low  Normal
## Dead at Year 1  21054   14442
## Alive at Year 1 27126 3804294
chi$expected  # the expected counts under the null hypothesis
##                        Low     Normal
## Dead at Year 1    442.2639   35053.74
## Alive at Year 1 47737.7361 3783682.26
chi$residuals # the Pearson residuals, (observed - expected) / sqrt(expected).
##                       Low     Normal
## Dead at Year 1  980.10778 -110.08988
## Alive at Year 1 -94.33735   10.59637
chi$stdres    # standardized residuals, (observed - expected) / sqrt(V)
##                       Low    Normal
## Dead at Year 1   990.8294 -990.8294
## Alive at Year 1 -990.8294  990.8294

2.3 pchisq

  • Density, distribution function, quantile function and random generation for the chi-squared (\(\chi^2\)) distribution with df degrees of freedom and optional non-centrality parameter ncp.
pchisq(q, df, ncp = 0, lower.tail = TRUE, log.p = FALSE)

  • q: (vector of) quantile(s).

  • df: degrees of freedom (non-negative, but can be non-integer).

  • lower.tail: logical; if TRUE (default), probabilities are \(P[X \le x]\), otherwise, \(P[X > x]\).

2.3.1 Example

To find the p-value that corresponds with a \(\chi^2\) test statistic of 7 from a test with one degree of freedom:

pchisq(7, df=1, lower.tail = FALSE)
## [1] 0.008150972

  • The P-value is the probability of observing a sample statistic as extreme as the test statistic – the area to the left of the red line – “upper” tail

  • We always set lower.tail = FALSE when calculating P-value of \(\chi^2\) test.

  • Another method to get the p-value

# default: lower.tail = TURE
1 - pchisq(7, df=1)
## [1] 0.008150972