Chapter05to07 {DanielBiostatistics10th} R Documentation

## Chapter 5, 6 and 7

### Description

Functions for Chapter 5, Some Important Sampling Distributions, Chapter 6, Estimation and Chapter 7, Hypothesis Testing.

### Usage

aggregated_z(
xbar,
n,
sd,
null.value,
alternative = c("two.sided", "less", "greater"),
conf.level = 0.95,
...
)

aggregated_t(
xbar,
xsd,
n,
null.value,
var.equal = FALSE,
alternative = c("two.sided", "less", "greater"),
conf.level = 0.95,
...
)

prop_CLT(
x,
n,
bool_obs,
xbar = x/n,
null.value,
alternative = c("two.sided", "less", "greater"),
conf.level = 0.95,
...
)

aggregated_var(
xsd,
n,
null.value,
alternative = c("two.sided", "less", "greater"),
conf.level = 0.95,
...
)


### Arguments

 xbar numeric scalar or length-two vector. Sample mean(s) for numeric variable(s) \bar{x} or (\bar{x}_1, \bar{x}_2). Sample proportion(s) for binary (i.e., logical) variable(s) \hat{p} or (\hat{p}_1, \hat{p}_2). In the case of two-sample tests, this could also be a numeric scalar indicating the difference in sample means \bar{x}_1-\bar{x}_2 or sample proportions \hat{p}_1-\hat{p}_2 n integer scalar n or length-two vector. Sample size(s) n or (n_1, n_2) sd numeric scalar or length-two vector. population standard deviation(s) \sigma or (\sigma_1, \sigma_2) null.value (optional) numeric scalar or length-two vector. Null value(s) of the population mean(s) (\mu_0, (\mu_{10}, \mu_{20}), or \mu_{10}-\mu_{20}) for functions aggregated_z() and aggregated_t(). Null value(s) of the population proportion(s) (p_0, (p_{10}, p_{20}), or p_{10}-p_{20}) for prop_CLT(). Null value(s) of the population variance(s) (ratio) (\sigma^2_0, (\sigma^2_{10}, \sigma^2_{20}), or \sigma^2_{10}/\sigma^2_{20}) for function aggregated_var(). If missing, only the confidence intervals will be computed. alternative character scalar, alternative hypothesis, either 'two.sided' (default), 'greater' or 'less' conf.level numeric scalar, confidence level, default 0.95 ... potential arguments, not in use currently xsd numeric scalar or length-two vector. Sample standard deviation(s) \sigma_{\bar{x}} or (\sigma_{\bar{x}_1}, \sigma_{\bar{x}_2}) var.equal logical scalar, whether to treat the two population variances as being equal (default FALSE) in function aggregated_t() x integer scalar or length-two vector, number of positive count(s) of binary (i.e., logical) variable(s) bool_obs logical vector of Boolean observations, used in one-sample z-test on proportion

### Details

Function aggregated_z() performs one- or two-sample z-test using the aggregated statistics of sample mean(s) and sample size(s) when null.value is provided. Otherwise, only the confidence interval based on z-distribution is computed.

Function aggregated_t() performs one- or two-sample t-test using the aggregated statistics of sample mean(s), sample standard deviation(s) and sample size(s) when null.value is provided. Otherwise, only the confidence interval based on t-distribution is computed.

Function prop_CLT() performs one- or two-sample z-test on proportion(s), using Central Limit Theorem when null.value is provided. Otherwise, only the confidence interval based on z-distribution is computed.

Function aggregated_var() performs one-sample \chi^2-test on variance, or two-sample F-test on variances, using the aggregated statistics of sample standard deviation(s) and sample size(s) when null.value is provided. Otherwise, only the confidence interval based on \chi^2- or F-distribution is computed.

### Value

Function aggregated_z() returns an 'htest' object when null.value is provided, otherwise returns a length-two numeric vector.

Function aggregated_t() returns an htest object when null.value is provided, otherwise returns a length-two numeric vector.

Function prop_CLT() returns an htest object when null.value is provided, otherwise returns a length-two numeric vector.

Function aggregated_var() returns an htest object when null.value is provided, otherwise returns a length-two numeric vector.

### References

Wayne W. Daniel, Biostatistics: A Foundation for Analysis in the Health Sciences, Tenth Edition. Wiley, ISBN: 978-1-119-62550-6.

### Examples

library(DanielBiostatistics10th)

# Page 142, Example 5.3.2
aggregated_z(xbar = 190, sd = 12.7, n = 10L, null.value = 185.6, alternative = 'greater')
# Page 143, Example 5.3.3
pnorm(125, mean = 120, sd = 15/sqrt(50)) - pnorm(115, mean = 120, sd = 15/sqrt(50))
aggregated_z(125, sd = 15, n = 50L, null.value = 120, alternative = 'less')$p.value - aggregated_z(115, sd = 15, n = 50L, null.value = 120, alternative = 'less')$p.value

# Page 145, Example 5.4.1
aggregated_z(xbar = c(92, 105), sd = 20, n = 15L, null.value = 0, alternative = 'less')
# Page 148, Example 5.4.2
aggregated_z(xbar = 20, sd = c(15, 20), n = c(35L, 40L), null.value = c(45, 30),
alternative = 'greater')

# Page 150, Example 5.5.1
prop_CLT(xbar = .4, n = 150L, null.value = .357, alternative = 'greater')
# Page 152, Example 5.5.2
prop_CLT(xbar = .45, n = 200L, null.value = .51, alternative = 'less')

# Page 155, Example 5.6.1
prop_CLT(xbar = .1, null.value = c(.28, .21), n = c(100L, 100L), alternative = 'greater')
# Page 155, Example 5.6.2
prop_CLT(xbar = .05, null.value = c(.34, .26), n = c(250L, 200L), alternative = 'less')

# Page 166, Example 6.2.1
aggregated_z(xbar = 22, n = 10L, sd = sqrt(45))
# Page 168, Example 6.2.2
aggregated_z(xbar = 84.3, n = 15L, sd = sqrt(144), conf.level = .99)
# Page 168, Example 6.2.3
aggregated_z(xbar = 17.2, n = 35L, sd = 8, conf.level = .9)
# Page 169, Example 6.2.4
aggregated_z(xbar = mean(EXA_C06_S02_04$ACTIVITY), n = nrow(EXA_C06_S02_04), sd = sqrt(.36)) # Page 173, Example 6.3.1 aggregated_t(xbar = 250.8, xsd = 130.9, n = 19L) # Page 177, Example 6.4.1 aggregated_z(xbar = c(4.5, 3.4), sd = sqrt(c(1, 1.5)), n = c(12L, 15L)) # Page 178, Example 6.4.2 aggregated_z(xbar = c(4.3, 13), sd = c(5.22, 8.97), n = c(328L, 64L), conf.level = .99) # Page 180, Example 6.4.3 aggregated_t(xbar = c(4.7, 8.8), xsd = c(9.3, 11.5), n = c(18L, 10L), var.equal = TRUE) # Page 181, Example 6.4.4 aggregated_t(xbar = c(4.7, 8.8), xsd = c(9.3, 11.5), n = c(18L, 10L)) # Welch slightly different from Cochran; textbook explained on Page 182 # Page 185, Example 6.5.1 prop_CLT(xbar = .18, n = 1220L) # Page 187, Example 6.6.1 prop_CLT(x = c(31L, 53L), n = c(68L, 255L), conf.level = .99) # Page 190, Example 6.7.1 n_671 = uniroot(f = function(n, sd, level = .95) { qnorm(1-(1-level)/2) * sd/sqrt(n) - 5 # half-width of CI <= 5 grams }, interval = c(0, 2e2), sd = 20) sprintf('Example 6.7.1 requires a sample size of %d.', ceiling(n_671$root))

# Page 192, Example 6.8.1
n_681 = uniroot(f = function(n, p, level = .95) {
qnorm(1-(1-level)/2) * sqrt(p*(1-p)/n) - .05
}, interval = c(0, 1e3), p = .35)
sprintf('Example 6.8.1 requires a sample size of %d.', ceiling(n_681$root)) # Page 196, Example 6.9.1 d691 = c(9.7, 12.3, 11.2, 5.1, 24.8, 14.8, 17.7) sqrt(aggregated_var(xsd = sd(d691), n = length(d691))) # Page 200, Example 6.10.1 aggregated_var(xsd = c(8.1, 5.9), n = c(16L, 4L)) # Page 222, Example 7.2.1 aggregated_z(xbar = 27, sd = sqrt(20), n = 10L, null.value = 30) # Page 226, Example 7.2.2 aggregated_z(xbar = 27, sd = sqrt(20), n = 10L, null.value = 30, alternative = 'less') # Page 228, Example 7.2.3 head(EXA_C07_S02_03) t.test(EXA_C07_S02_03$DAYS, mu = 15)
# Page 231, Example 7.2.4
aggregated_z(xbar = 146, sd = 27, n = 157L, null.value = 140, alternative = 'greater')
# Page 232, Example 7.2.5
d725 = c(33.38, 32.15, 34.34, 33.95, 33.46, 34.13, 33.99, 34.10, 33.85,
34.23, 34.45, 34.19, 33.97, 32.73, 34.05)
t.test(d725, mu = 34.5)

# Page 237, Example 7.3.1
aggregated_z(xbar = c(4.5, 3.4), sd = sqrt(c(1, 1.5)), n = c(12L, 15L), null.value = 0)
# Page 239, Example 7.3.2
with(EXA_C07_S03_02, t.test(x = CONTROL, y = SCI, alternative = 'less', var.equal = TRUE))
# Page 240, Example 7.3.3
aggregated_t(xbar = c(19.16, 9.53), xsd = c(5.29, 2.69), n = c(15L, 30L), null.value = 0)
# Page 242, Example 7.3.4
aggregated_z(xbar = c(59.01, 46.61), sd = c(44.89, 34.85), n = c(53L, 54L), null.value = 0,
alternative = 'greater')

# Page 251, Example 7.4.1
with(EXA_C07_S04_01, t.test(x = POSTOP, y = PREOP, alternative = 'greater', paired = TRUE))

# Page 258, Example 7.5.1
prop_CLT(x = 24L, n = 301L, null.value = .063, alternative = 'greater')

# Page 261, Example 7.6.1
prop_CLT(x = c(24L, 11L), n = c(44L, 29L), null.value = 0, alternative = 'greater')

# Page 264, Example 7.7.1