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Last data update: 2014.03.03

R: Summarized t-test

tsum.test

R Documentation

Summarized t-test

Description

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.

Usage

tsum.test(mean.x, s.x = NULL, n.x = NULL, mean.y = NULL, s.y = NULL,
n.y = NULL, alternative = "two.sided", mu = 0, var.equal = FALSE,
conf.level = 0.95)

Arguments

`mean.x`	a single number representing the sample mean of `x`
`s.x`	a single number representing the sample standard deviation for `x`
`n.x`	a single number representing the sample size for `x`
`mean.y`	a single number representing the sample mean of `y`
`s.y`	a single number representing the sample standard deviation for `y`
`n.y`	a single number representing the sample size for `y`
`alternative`	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`.
`mu`	is a single number representing the value of the mean or difference in means specified by the null hypothesis.
`var.equal`	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.
`conf.level`	is the confidence level for the returned confidence interval; it must lie between zero and one.

Details

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.

Value

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`	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`	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.
`alternative`	records the value of the input argument alternative: `"greater"` , `"less"` or `"two.sided"`.
`data.name`	a character string (vector of length 1) containing the names x and y for the two summarized samples.

Null Hypothesis

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").

Test Assumptions

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.

Confidence Intervals

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.

Author(s)

Alan T. Arnholt

References

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.

Examples

tsum.test(mean.x=5.6, s.x=2.1, n.x=16, mu=4.9, alternative="greater")
        # Problem 6.31 on page 324 of BSDA states:  The chamber of commerce
        # of a particular city claims that the mean carbon dioxide
        # level of air polution is no greater than 4.9 ppm.  A random
        # sample of 16 readings resulted in a sample mean of 5.6 ppm,
        # and s=2.1 ppm.  One-sided one-sample t-test.  The null 
        # hypothesis is that the population mean for 'x' is 4.9.   
        # The alternative hypothesis states that it is greater than 4.9.  

x <- rnorm(12) 
tsum.test(mean(x), sd(x), n.x=12)
        # Two-sided one-sample t-test. The null hypothesis is that  
        # the population mean for 'x' is zero. The alternative 
        # hypothesis states  that it is either greater or less 
        # than zero. A confidence interval for the population mean 
        # will be computed.  Note: above returns same answer as: 
t.test(x)
   
x <- c(7.8, 6.6, 6.5, 7.4, 7.3, 7.0, 6.4, 7.1, 6.7, 7.6, 6.8) 
y <- c(4.5, 5.4, 6.1, 6.1, 5.4, 5.0, 4.1, 5.5) 
tsum.test(mean(x), s.x=sd(x), n.x=11 ,mean(y), s.y=sd(y), n.y=8, mu=2)
        # Two-sided standard two-sample t-test.  The null hypothesis  
        # is that the population mean for 'x' less that for 'y' is 2. 
        # The alternative hypothesis is that this difference is not 2. 
        # A confidence interval for the true difference will be computed.
        # Note: above returns same answer as: 
t.test(x, y)
        
tsum.test(mean(x), s.x=sd(x), n.x=11, mean(y), s.y=sd(y), n.y=8, conf.level=0.90)
        # Two-sided standard two-sample t-test.  The null hypothesis 
        # is that the population mean for 'x' less that for 'y' is zero.  
        # The alternative hypothesis is that this difference is not
        # zero.  A 90% confidence interval for the true difference will 
        # be computed.  Note: above returns same answer as:
t.test(x, y, conf.level=0.90)

Results


R version 3.3.1 (2016-06-21) -- "Bug in Your Hair"
Copyright (C) 2016 The R Foundation for Statistical Computing
Platform: x86_64-pc-linux-gnu (64-bit)

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Type 'demo()' for some demos, 'help()' for on-line help, or
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Type 'q()' to quit R.

> library(BSDA)
Loading required package: e1071
Loading required package: lattice

Attaching package: 'BSDA'

The following object is masked from 'package:datasets':

    Orange

> png(filename="/home/ddbj/snapshot/RGM3/R_CC/result/BSDA/tsum.test.Rd_%03d_medium.png", width=480, height=480)
> ### Name: tsum.test
> ### Title: Summarized t-test
> ### Aliases: tsum.test
> ### Keywords: htest
> 
> ### ** Examples
> 
> tsum.test(mean.x=5.6, s.x=2.1, n.x=16, mu=4.9, alternative="greater")

	One-sample t-Test

data:  Summarized x
t = 1.3333, df = 15, p-value = 0.1012
alternative hypothesis: true mean is greater than 4.9
95 percent confidence interval:
 4.679649       NA
sample estimates:
mean of x 
      5.6 

Warning message:
In tsum.test(mean.x = 5.6, s.x = 2.1, n.x = 16, mu = 4.9, alternative = "greater") :
  argument 'var.equal' ignored for one-sample test.
>         # Problem 6.31 on page 324 of BSDA states:  The chamber of commerce
>         # of a particular city claims that the mean carbon dioxide
>         # level of air polution is no greater than 4.9 ppm.  A random
>         # sample of 16 readings resulted in a sample mean of 5.6 ppm,
>         # and s=2.1 ppm.  One-sided one-sample t-test.  The null 
>         # hypothesis is that the population mean for 'x' is 4.9.   
>         # The alternative hypothesis states that it is greater than 4.9.  
> 
> x <- rnorm(12) 
> tsum.test(mean(x), sd(x), n.x=12)

	One-sample t-Test

data:  Summarized x
t = 0.8864, df = 11, p-value = 0.3944
alternative hypothesis: true mean is not equal to 0
95 percent confidence interval:
 -0.3244271  0.7619346
sample estimates:
mean of x 
0.2187537 

Warning message:
In tsum.test(mean(x), sd(x), n.x = 12) :
  argument 'var.equal' ignored for one-sample test.
>         # Two-sided one-sample t-test. The null hypothesis is that  
>         # the population mean for 'x' is zero. The alternative 
>         # hypothesis states  that it is either greater or less 
>         # than zero. A confidence interval for the population mean 
>         # will be computed.  Note: above returns same answer as: 
> t.test(x)

	One Sample t-test

data:  x
t = 0.8864, df = 11, p-value = 0.3944
alternative hypothesis: true mean is not equal to 0
95 percent confidence interval:
 -0.3244271  0.7619346
sample estimates:
mean of x 
0.2187537 

>    
> x <- c(7.8, 6.6, 6.5, 7.4, 7.3, 7.0, 6.4, 7.1, 6.7, 7.6, 6.8) 
> y <- c(4.5, 5.4, 6.1, 6.1, 5.4, 5.0, 4.1, 5.5) 
> tsum.test(mean(x), s.x=sd(x), n.x=11 ,mean(y), s.y=sd(y), n.y=8, mu=2)

	Welch Modified Two-Sample t-Test

data:  Summarized x and y
t = -0.85278, df = 11.303, p-value = 0.4115
alternative hypothesis: true difference in means is not equal to 2
95 percent confidence interval:
 1.127160 2.384203
sample estimates:
mean of x mean of y 
 7.018182  5.262500 

>         # Two-sided standard two-sample t-test.  The null hypothesis  
>         # is that the population mean for 'x' less that for 'y' is 2. 
>         # The alternative hypothesis is that this difference is not 2. 
>         # A confidence interval for the true difference will be computed.
>         # Note: above returns same answer as: 
> t.test(x, y)

	Welch Two Sample t-test

data:  x and y
t = 6.1281, df = 11.303, p-value = 6.617e-05
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 1.127160 2.384203
sample estimates:
mean of x mean of y 
 7.018182  5.262500 

>         
> tsum.test(mean(x), s.x=sd(x), n.x=11, mean(y), s.y=sd(y), n.y=8, conf.level=0.90)

	Welch Modified Two-Sample t-Test

data:  Summarized x and y
t = 6.1281, df = 11.303, p-value = 6.617e-05
alternative hypothesis: true difference in means is not equal to 0
90 percent confidence interval:
 1.242424 2.268940
sample estimates:
mean of x mean of y 
 7.018182  5.262500 

>         # Two-sided standard two-sample t-test.  The null hypothesis 
>         # is that the population mean for 'x' less that for 'y' is zero.  
>         # The alternative hypothesis is that this difference is not
>         # zero.  A 90% confidence interval for the true difference will 
>         # be computed.  Note: above returns same answer as:
> t.test(x, y, conf.level=0.90)

	Welch Two Sample t-test

data:  x and y
t = 6.1281, df = 11.303, p-value = 6.617e-05
alternative hypothesis: true difference in means is not equal to 0
90 percent confidence interval:
 1.242424 2.268940
sample estimates:
mean of x mean of y 
 7.018182  5.262500 

> 
> 
> 
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> 
> dev.off()
null device 
          1 
>