| t.test {stats} | R Documentation |
Performs one and two sample t-tests on vectors of data.\n", "
\n", "\n", "\n", "\n",
"t.test(x, ...)\n",
"\n",
"## Default S3 method:\n",
"t.test(x, y = NULL,\n",
" alternative = c(\"two.sided\", \"less\", \"greater\"),\n",
" mu = 0, paired = FALSE, var.equal = FALSE,\n",
" conf.level = 0.95, ...)\n",
"\n",
"## S3 method for class 'formula'\n",
"t.test(formula, data, subset, na.action, ...)\n",
"\n",
"\n",
"\n",
"x | \n",
"\n",
" a (non-empty) numeric vector of data values. \n", " |
y | \n",
"\n",
" an optional (non-empty) numeric vector of data values. \n", " |
alternative | \n",
"\n",
" a character string specifying the alternative\n",
"hypothesis, must be one of |
mu | \n",
"\n",
" a number indicating the true value of the mean (or\n", "difference in means if you are performing a two sample test). \n", " |
paired | \n",
"\n",
" a logical indicating whether you want a paired\n", "t-test. \n", " |
var.equal | \n",
"\n",
" a logical variable indicating whether to treat the\n",
"two variances as being equal. If |
conf.level | \n",
"\n",
" confidence level of the interval. \n", " |
formula | \n",
"\n",
" a formula of the form |
data | \n",
"\n",
" an optional matrix or data frame (or similar: see\n",
" |
subset | \n",
"\n",
" an optional vector specifying a subset of observations\n", "to be used. \n", " |
na.action | \n",
"\n",
" a function which indicates what should happen when\n",
"the data contain |
... | \n",
"\n",
" further arguments to be passed to or from methods. \n", " |
The formula interface is only applicable for the 2-sample tests.\n", "
\n", "alternative = \"greater\" is the alternative that x has a\n",
"larger mean than y.\n",
"
If paired is TRUE then both x and y must\n",
"be specified and they must be the same length. Missing values are\n",
"silently removed (in pairs if paired is TRUE). If\n",
"var.equal is TRUE then the pooled estimate of the\n",
"variance is used. By default, if var.equal is FALSE\n",
"then the variance is estimated separately for both groups and the\n",
"Welch modification to the degrees of freedom is used.\n",
"
If the input data are effectively constant (compared to the larger of the\n", "two means) an error is generated.\n", "
\n", "\n", "\n", "A list with class \"htest\" containing the following components:\n",
"
statistic | \n",
"\n",
" the value of the t-statistic. \n", " |
parameter | \n",
"\n",
" the degrees of freedom for the t-statistic. \n", " |
p.value | \n",
"\n",
" the p-value for the test. \n", " |
conf.int | \n",
"\n",
" a confidence interval for the mean appropriate to the\n", "specified alternative hypothesis. \n", " |
estimate | \n",
"\n",
" the estimated mean or difference in means depending on\n", "whether it was a one-sample test or a two-sample test. \n", " |
null.value | \n",
"\n",
" the specified hypothesized value of the mean or mean\n", "difference depending on whether it was a one-sample test or a\n", "two-sample test. \n", " |
alternative | \n",
"\n",
" a character string describing the alternative\n", "hypothesis. \n", " |
method | \n",
"\n",
" a character string indicating what type of t-test was\n", "performed. \n", " |
data.name | \n",
"\n",
" a character string giving the name(s) of the data. \n", " |
prop.test\n",
"
\n",
"require(graphics)\n",
"\n",
"t.test(1:10, y = c(7:20)) # P = .00001855\n",
"t.test(1:10, y = c(7:20, 200)) # P = .1245 -- NOT significant anymore\n",
"\n",
"## Classical example: Student's sleep data\n",
"plot(extra ~ group, data = sleep)\n",
"## Traditional interface\n",
"with(sleep, t.test(extra[group == 1], extra[group == 2]))\n",
"## Formula interface\n",
"t.test(extra ~ group, data = sleep)\n",
"\n",
"\n",
"| wilcox.test {stats} | R Documentation |
Performs one- and two-sample Wilcoxon tests on vectors of data; the\n", "latter is also known as ‘Mann-Whitney’ test.\n", "
\n", "\n", "\n", "\n",
"wilcox.test(x, ...)\n",
"\n",
"## Default S3 method:\n",
"wilcox.test(x, y = NULL,\n",
" alternative = c(\"two.sided\", \"less\", \"greater\"),\n",
" mu = 0, paired = FALSE, exact = NULL, correct = TRUE,\n",
" conf.int = FALSE, conf.level = 0.95, ...)\n",
"\n",
"## S3 method for class 'formula'\n",
"wilcox.test(formula, data, subset, na.action, ...)\n",
"\n",
"\n",
"\n",
"x | \n",
"\n",
" numeric vector of data values. Non-finite (e.g., infinite or\n", "missing) values will be omitted. \n", " |
y | \n",
"\n",
" an optional numeric vector of data values: as with |
alternative | \n",
"\n",
" a character string specifying the alternative\n",
"hypothesis, must be one of |
mu | \n",
"\n",
" a number specifying an optional parameter used to form the\n", "null hypothesis. See ‘Details’. \n", " |
paired | \n",
"\n",
" a logical indicating whether you want a paired test. \n", " |
exact | \n",
"\n",
" a logical indicating whether an exact p-value\n", "should be computed. \n", " |
correct | \n",
"\n",
" a logical indicating whether to apply continuity\n", "correction in the normal approximation for the p-value. \n", " |
conf.int | \n",
"\n",
" a logical indicating whether a confidence interval\n", "should be computed. \n", " |
conf.level | \n",
"\n",
" confidence level of the interval. \n", " |
formula | \n",
"\n",
" a formula of the form |
data | \n",
"\n",
" an optional matrix or data frame (or similar: see\n",
" |
subset | \n",
"\n",
" an optional vector specifying a subset of observations\n", "to be used. \n", " |
na.action | \n",
"\n",
" a function which indicates what should happen when\n",
"the data contain |
... | \n",
"\n",
" further arguments to be passed to or from methods. \n", " |
The formula interface is only applicable for the 2-sample tests.\n", "
\n", "If only x is given, or if both x and y are given\n",
"and paired is TRUE, a Wilcoxon signed rank test of the\n",
"null that the distribution of x (in the one sample case) or of\n",
"x - y (in the paired two sample case) is symmetric about\n",
"mu is performed.\n",
"
Otherwise, if both x and y are given and paired\n",
"is FALSE, a Wilcoxon rank sum test (equivalent to the\n",
"Mann-Whitney test: see the Note) is carried out. In this case, the\n",
"null hypothesis is that the distributions of x and y\n",
"differ by a location shift of mu and the alternative is that\n",
"they differ by some other location shift (and the one-sided\n",
"alternative \"greater\" is that x is shifted to the right\n",
"of y).\n",
"
By default (if exact is not specified), an exact p-value\n",
"is computed if the samples contain less than 50 finite values and\n",
"there are no ties. Otherwise, a normal approximation is used.\n",
"
Optionally (if argument conf.int is true), a nonparametric\n",
"confidence interval and an estimator for the pseudomedian (one-sample\n",
"case) or for the difference of the location parameters x-y is\n",
"computed. (The pseudomedian of a distribution F is the median\n",
"of the distribution of (u+v)/2, where u and v are\n",
"independent, each with distribution F. If F is symmetric,\n",
"then the pseudomedian and median coincide. See Hollander & Wolfe\n",
"(1973), page 34.) Note that in the two-sample case the estimator for\n",
"the difference in location parameters does not estimate the\n",
"difference in medians (a common misconception) but rather the median\n",
"of the difference between a sample from x and a sample from\n",
"y.\n",
"
If exact p-values are available, an exact confidence interval is\n",
"obtained by the algorithm described in Bauer (1972), and the\n",
"Hodges-Lehmann estimator is employed. Otherwise, the returned\n",
"confidence interval and point estimate are based on normal\n",
"approximations. These are continuity-corrected for the interval but\n",
"not the estimate (as the correction depends on the\n",
"alternative).\n",
"
With small samples it may not be possible to achieve very high\n", "confidence interval coverages. If this happens a warning will be given\n", "and an interval with lower coverage will be substituted.\n", "
\n", "\n", "\n", "A list with class \"htest\" containing the following components:\n",
"
statistic | \n",
"\n",
" the value of the test statistic with a name\n", "describing it. \n", " |
parameter | \n",
"\n",
" the parameter(s) for the exact distribution of the\n", "test statistic. \n", " |
p.value | \n",
"\n",
" the p-value for the test. \n", " |
null.value | \n",
"\n",
" the location parameter |
alternative | \n",
"\n",
" a character string describing the alternative\n", "hypothesis. \n", " |
method | \n",
"\n",
" the type of test applied. \n", " |
data.name | \n",
"\n",
" a character string giving the names of the data. \n", " |
conf.int | \n",
"\n",
" a confidence interval for the location parameter.\n",
"(Only present if argument |
estimate | \n",
"\n",
" an estimate of the location parameter.\n",
"(Only present if argument |
This function can use large amounts of memory and stack (and even\n",
"crash R if the stack limit is exceeded) if exact = TRUE and\n",
"one sample is large (several thousands or more).\n",
"
The literature is not unanimous about the definitions of the Wilcoxon\n", "rank sum and Mann-Whitney tests. The two most common definitions\n", "correspond to the sum of the ranks of the first sample with the\n", "minimum value subtracted or not: R subtracts and S-PLUS does not,\n", "giving a value which is larger by m(m+1)/2 for a first sample\n", "of size m. (It seems Wilcoxon's original paper used the\n", "unadjusted sum of the ranks but subsequent tables subtracted the\n", "minimum.)\n", "
\n", "R's value can also be computed as the number of all pairs\n",
"(x[i], y[j]) for which y[j] is not greater than\n",
"x[i], the most common definition of the Mann-Whitney test.\n",
"
David F. Bauer (1972),\n", "Constructing confidence sets using rank statistics.\n", "Journal of the American Statistical Association\n", "67, 687–690.\n", "
\n", "Myles Hollander and Douglas A. Wolfe (1973),\n",
"Nonparametric Statistical Methods.\n",
"New York: John Wiley & Sons.\n",
"Pages 27–33 (one-sample), 68–75 (two-sample).
\n",
"Or second edition (1999).\n",
"
psignrank, pwilcox.\n",
"
wilcox_test in package\n",
"coin for exact, asymptotic and Monte Carlo\n",
"conditional p-values, including in the presence of ties.\n",
"
kruskal.test for testing homogeneity in location\n",
"parameters in the case of two or more samples;\n",
"t.test for an alternative under normality\n",
"assumptions [or large samples]\n",
"
\n",
"require(graphics)\n",
"## One-sample test.\n",
"## Hollander & Wolfe (1973), 29f.\n",
"## Hamilton depression scale factor measurements in 9 patients with\n",
"## mixed anxiety and depression, taken at the first (x) and second\n",
"## (y) visit after initiation of a therapy (administration of a\n",
"## tranquilizer).\n",
"x <- c(1.83, 0.50, 1.62, 2.48, 1.68, 1.88, 1.55, 3.06, 1.30)\n",
"y <- c(0.878, 0.647, 0.598, 2.05, 1.06, 1.29, 1.06, 3.14, 1.29)\n",
"wilcox.test(x, y, paired = TRUE, alternative = \"greater\")\n",
"wilcox.test(y - x, alternative = \"less\") # The same.\n",
"wilcox.test(y - x, alternative = \"less\",\n",
" exact = FALSE, correct = FALSE) # H&W large sample\n",
" # approximation\n",
"\n",
"## Two-sample test.\n",
"## Hollander & Wolfe (1973), 69f.\n",
"## Permeability constants of the human chorioamnion (a placental\n",
"## membrane) at term (x) and between 12 to 26 weeks gestational\n",
"## age (y). The alternative of interest is greater permeability\n",
"## of the human chorioamnion for the term pregnancy.\n",
"x <- c(0.80, 0.83, 1.89, 1.04, 1.45, 1.38, 1.91, 1.64, 0.73, 1.46)\n",
"y <- c(1.15, 0.88, 0.90, 0.74, 1.21)\n",
"wilcox.test(x, y, alternative = \"g\") # greater\n",
"wilcox.test(x, y, alternative = \"greater\",\n",
" exact = FALSE, correct = FALSE) # H&W large sample\n",
" # approximation\n",
"\n",
"wilcox.test(rnorm(10), rnorm(10, 2), conf.int = TRUE)\n",
"\n",
"## Formula interface.\n",
"boxplot(Ozone ~ Month, data = airquality)\n",
"wilcox.test(Ozone ~ Month, data = airquality,\n",
" subset = Month %in% c(5, 8))\n",
"\n",
"\n",
"