t.test
Student's t-test

### Description

Performs a one-sample, two-sample, or paired t-tests.

### Usage

    t.test(x, ...)
    ## Default S3 method:
    t.test(x, y = NULL, alternative = c("two.sided", "less", "greater"),
        mu = 0, paired = FALSE, var.equal = FALSE, conf.level = 0.95,
        treatment = NULL, ...)
    ## S3 method for class 'formula':
    t.test(formula, data, subset, na.action, ...)

### Arguments

x | a numeric vector that contains the sample values. Missing values (NAs) and infinite values (Infs) are removed before calculation.
y | a numeric vector that contains the sample values. Missing values (NAs) and infinite values (Infs) are removed before calculation. If paired = TRUE, length(x) must equal length(y) and an observation pair (x[i], y[i]) is removed if it has at least one NA or Inf value.
alternative | a character string that specifies the alternative hypothesis. Options are: "two.sided": the true mean is not equal to mu, "greater": the true mean is greater than mu, "less": the true mean is less than mu. You only need to enter enough of the character string to create a unique match. Depending on the type of test, alternative refers to one of the following: One-sample and paired t-tests: The true mean of the parent population in relation to the hypothesized value mu. Two-sample t-tests: The difference between the true population mean for x and that for y, in relation to mu.
mu | a numeric scalar that represents the value of the mean or the difference of means specified by the null hypothesis.
paired | a logical value. If TRUE, length(x) must equal length(y) and the values in x and y are treated as observation pairs (x[i], y[i]).
var.equal | a logical value. If TRUE, the function is evaluated under the assumption that the variances of the parent populations of x and y are equal. This argument only gets used for two-sample (i.e., unpaired) tests.
conf.level | a numeric value in the range [0, 1] that specifies the confidence level for the returned confidence interval.
treatment | a vector of any kind with exactly two unique values and the same length as x. If supplied, it is used to split x into two samples and y is not used. The t-statistic numerator is the difference between the means of the two groups. This argument is not in R.
formula | a formula of the form v~g that gives the name of a numeric variable (v) and the name of a grouping factor (g). g must have exactly two levels and length equal to that of v.
data | a data frame that contains the variables named in the formula and subset arguments. Defaults to the parent frame from which the function was called.
subset | a vector that specifies which subset of the rows of the data should be used. This can be a logical vector that is replicated to have length equal to the number of rows of data, a numeric vector that indicates the row numbers to be included, or a character vector of the row names that should be included. By default, all rows are included.
na.action | a function that handles missing values. See na.action for details.
... | additional arguments.

### Details

If a value for y is supplied then the value of paired is used. If paired == TRUE, a paired t-test is computed. If paired == FALSE, a two-sample t-test is computed and the value of var.equal is used. If var.equal == TRUE then the standard two-sample t-test is computed. If var.equal == FALSE then the Welsh modified two-sample t-test is computed.
One-sample t-test

statistic: t = (mean(x)-mu) / (sqrt(var(x))/sqrt(length(x)))

If x was drawn from a normal population, t has a t-distribution with length(x)-1 degrees of freedom under the null hypothesis. The null hypothesis in this case is that the mean of the population from which x is drawn is mu.
Paired t-test

statistic: t = (mean(d)-mu) / (sqrt(var(d))/sqrt(length(d)))
where d is the vector of differences x-y

Under the null hypothesis, t follows a t-distribution with length(d)-1 degrees of freedom, assuming normality of the differences d. The null hypothesis in this case is that the population mean of the difference x-y is equal to mu.
Equal-variance two-sample t-test

statistic: t = (mean(x)-mean(y)-mu) / s1
where:
s1 = sp*sqrt(1/nx+1/ny)
sp = sqrt(((nx-1)*var(x)+(ny-1)*var(y))/(nx+ny-2))
nx = length(x)
ny = length(y)

Assuming that x and y come from normal populations with equal variances, t has a t-distribution with nx+ny-2 degrees of freedom under the null hypothesis. The null hypothesis in this case is that the population mean for x minus that for y is mu.
Welch modified two-sample t-test

statistic: t = (mean(x)-mean(y)-mu)/s2
where:
s2 = sqrt(var(x)/nx+var(y)/ny)
nx = length(x)
ny = length(y)

If x and y come from normal populations, the distribution of t under the null hypothesis can be approximated by a t-distribution with (non-integral) degrees of freedom:

1/((c^2)/(nx-1)+((1-c)^2)/(ny-1))
where c = var(x)/(nx*s2^2)

The null hypothesis in this case is that the population mean for x minus 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, that is, "greater", "less", "two.sided".
In all cases, if the distributions are not normal but sample sizes are large, then t-distributions hold approximately (under certain regularity conditions). However, large sample sizes are no help if you use the pooled-variance test and the variances are not equal.
Value
a list of class "htest" that contains the following components:

statistic | the t-statistic with names attribute "t".
parameter | the degrees of freedom of the t-distribution associated with statistic with names attribute "df".
p.value | the p-value for the test.
conf.int | the confidence interval (vector of length 2) for the true mean or difference in means. The confidence level specified by the input argument conf.level is recorded as its conf.level attribute.
estimate | the sample mean(s) or mean of the differences (vector of length 1 or 2) that estimate the corresponding population parameters with a names attribute that describes the estimate, for example "mean of the differences".
null.value | the value of the mean or difference in means specified by the null hypothesis with a names attribute that describes the null.value, for example "difference in means". The value for null.value is equal to the value of the input argument mu.
alternative | a character string that returns the alternative hypothesis ("two.sided", "greater", or "less") as specified in the alternative argument.
method | a character string for the name of the test used in the calculation.
data.name | a character string (vector of length 1) that contains the names of the x and, if provided, y input vectors.

References
Box, G. E. P. (1953), "Non-normality and Tests on Variances," Biometrika, pp. 318-335.
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.
Differences between Spotfire Enterprise Runtime for R and Open-source R
The argument "treatment" is not in open-source R.
See Also
aov, wilcox.test
Examples

    # Two-sided one-sample t-test.
    # Null hypothesis is that the population mean for 'x' is zero.
    # Alternative hypothesis states that it is either greater or less than zero.
    # Computes a confidence interval for the population mean.
    x <- rnorm(12)
    t.test(x)
    # One-sided paired t-test.
    # Null hypothesis is that the population mean "before"
    # and the one "after" are the same, or equivalently that
    # the mean change ("after" minus "before") is zero.
    # Alternative hypothesis is that the mean "after" is less than the one "before",
    # or equivalently that the mean change is negative.
    # Computes a confidence interval for the mean change.
    before <- c(31, 20, 18, 17, 9, 8, 10, 7)
    after <- c(18, 17, 14, 11, 10, 7, 5, 6)
    t.test(after, before, alternative = "less", paired = TRUE)
    # Two-sided two-sample t-test.
    # Null hypothesis is that the population means for 'x' and 'y' are the same.
    # Alternative hypothesis is that they are not.
    # The confidence interval for the difference in true means ('x' minus 'y')
    # will have a confidence level of 0.90.
    x <- c(7.8, 6.6, 6.5, 7.4, 7.3, 7., 6.4, 7.1, 6.7, 7.6, 6.8)
    y <- c(4.5, 5.4, 6.1, 6.1, 5.4, 5., 4.1, 5.5)
    t.test(x, y, conf.level = 0.90)
    # Two-sided pooled-variance two-sample t-test.
    # This assumes that the two populations variances are equal.
    # Null hypothesis is that the population mean for 'x' minus that for 'y' is 2.
    # Alternative hypothesis is that this difference is not 2.
    # Computes a confidence interval for the true difference.
    t.test(x, y, mu = 2)
    # Formula interface
    t.test(Mileage ~ I(Type
        subset = (Type != "Sporty"))

Package stats version 6.1.5-18
Package Index