Performs a one-sample, two-sample, or a Welch modified two-sample t-test based on user supplied summary information. Output is identical to that produced with t.test
.
tsum.test(
mean.x,
s.x = NULL,
n.x = NULL,
mean.y = NULL,
s.y = NULL,
n.y = NULL,
alternative = c("two.sided", "less", "greater"),
mu = 0,
var.equal = FALSE,
conf.level = 0.95,
...
)
a single number representing the sample mean of x
a single number representing the sample standard deviation of x
a single number representing the sample size of x
a single number representing the sample mean of y
a single number representing the sample standard deviation of y
a single number representing the sample size of y
is a character string, one of "greater"
, "less"
, or "two.sided"
, or just the initial letter of each, indicating the specification of the alternative hypothesis. For one-sample tests, alternative
refers to the true mean of the parent population in relation to the hypothesized value mu
. For the standard two-sample tests, alternative
refers to the difference between the true population mean for x
and that for y
, in relation to mu
. For the one-sample and paired t-tests, alternative
refers to the true mean of the parent population in relation to the hypothesized value mu
. For the standard and Welch modified two-sample t-tests, alternative
refers to the difference between the true population mean for x
and that for y
, in relation to mu
. For the one-sample t-tests, alternative refers to the true mean of the parent population in relation to the hypothesized value mu
. For the standard and Welch modified two-sample t-tests, alternative refers to the difference between the true population mean for x
and that for y
, in relation to mu
.
is a single number representing the value of the mean or difference in means specified by the null hypothesis.
logical flag: if TRUE
, the variances of the parent populations of x
and y
are assumed equal. Argument var.equal
should be supplied only for the two-sample tests.
is the confidence level for the returned confidence interval; it must lie between zero and one.
Other arguments passed onto tsum.test()
A list of class htest
, containing the following components:
statistic
the t-statistic, with names attribute "t"
parameters
is the degrees of freedom of the t-distribution associated with statistic. Component parameters
has names attribute "df"
.
p.value
the p-value for the test
conf.int
is a confidence interval (vector of length 2) for the true mean or difference in means. The confidence level is recorded in the attribute conf.level
. When alternative is not "two.sided"
, the confidence interval will be half-infinite, to reflect the interpretation of a confidence interval as the set of all values k
for which one would not reject the null hypothesis that the true mean or difference in means is k
. Here infinity will be represented by Inf
.
estimate
is a vector of length 1 or 2, giving the sample mean(s) or mean of differences; these estimate the corresponding population parameters. Component estimate
has a names attribute describing its elements.
null.value
is the value of the mean or difference in means specified by the null hypothesis. This equals the input argument mu
. Component null.value
has a names attribute describing its elements.
records the value of the input argument alternative: "greater"
, "less"
or "two.sided"
.
is a character string (vector of length 1) containing the names x and y for the two summarized samples.
If y
is NULL
, a one-sample t-test is carried out with x
. If y
is not NULL
, either a standard or Welch modified two-sample t-test is performed, depending on whether var.equal
is TRUE
or FALSE
.
For the one-sample t-test, the null hypothesis is that the mean of the population from which x
is drawn is mu
. For the standard and Welch modified two-sample t-tests, the null hypothesis is that the population mean for x
less that for y
is mu
.
The alternative hypothesis in each case indicates the direction of divergence of the population mean for x
(or difference of means for x
and y
) from mu
(i.e., "greater"
, "less"
, or "two.sided"
).
The assumption of equal population variances is central to the standard two-sample t-test. This test can be misleading when population variances are not equal, as the null distribution of the test statistic is no longer a t-distribution. If the assumption of equal variances is doubtful with respect to a particular dataset, the Welch modification of the t-test should be used.
The t-test and the associated confidence interval are quite robust with respect to level toward heavy-tailed non-Gaussian distributions (e.g., data with outliers). However, the t-test is non-robust with respect to power, and the confidence interval is non-robust with respect to average length, toward these same types of distributions.
For each of the above tests, an expression for the related confidence interval (returned component conf.int
) can be obtained in the usual way by inverting the expression for the test statistic. Note that, as explained under the description of conf.int
, the confidence interval will be half-infinite when alternative is not "two.sided"
; infinity will be represented by Inf
.
Kitchens, L.J. 2003. Basic Statistics and Data Analysis. Duxbury.
Hogg, R. V. and Craig, A. T. 1970. Introduction to Mathematical Statistics, 3rd ed. Toronto, Canada: Macmillan.
Mood, A. M., Graybill, F. A. and Boes, D. C. 1974. Introduction to the Theory of Statistics, 3rd ed. New York: McGraw-Hill.
Snedecor, G. W. and Cochran, W. G. 1980. Statistical Methods, 7th ed. Ames, Iowa: Iowa State University Press.
# 95% Confidence Interval for mu1 - mu2, assuming equal variances
round(tsum.test(mean.x = 53/15, mean.y = 77/11, s.x=sqrt((222 - 15*(53/15)^2)/14),
s.y = sqrt((560 - 11*(77/11)^2)/10), n.x = 15, n.y = 11, var.equal = TRUE)$conf, 2)
#> [1] -4.72 -2.22
#> attr(,"conf.level")
#> [1] 0.95
# One Sample t-test
tsum.test(mean.x = 4, s.x = 2.89, n.x = 25, mu = 2.5)
#>
#> One Sample t-test
#>
#> data: User input summarized values for x
#> t = 2.5952, df = 24, p-value = 0.01588
#> alternative hypothesis: true mean is not equal to 2.5
#> 95 percent confidence interval:
#> 2.807067 5.192933
#> sample estimates:
#> mean of x
#> 4
#>