 Student's ttest

A ttest is any statistical hypothesis test in which the test statistic follows a Student's t distribution if the null hypothesis is supported. It is most commonly applied when the test statistic would follow a normal distribution if the value of a scaling term in the test statistic were known. When the scaling term is unknown and is replaced by an estimate based on the data, the test statistic (under certain conditions) follows a Student's t distribution.
Contents
History
The tstatistic was introduced in 1908 by William Sealy Gosset, a chemist working for the Guinness brewery in Dublin, Ireland ("Student" was his pen name).^{[1]}^{[2]}^{[3]} Gosset had been hired due to Claude Guinness's policy of recruiting the best graduates from Oxford and Cambridge to apply biochemistry and statistics to Guinness' industrial processes.^{[2]} Gosset devised the ttest as a way to cheaply monitor the quality of stout. He published the test in Biometrika in 1908, but was forced to use a pen name by his employer, who regarded the fact that they were using statistics as a trade secret. In fact, Gosset's identity was known to fellow statisticians.^{[4]}
Uses
Among the most frequently used ttests are:
 A onesample location test of whether the mean of a normally distributed population has a value specified in a null hypothesis.
 A two sample location test of the null hypothesis that the means of two normally distributed populations are equal. All such tests are usually called Student's ttests, though strictly speaking that name should only be used if the variances of the two populations are also assumed to be equal; the form of the test used when this assumption is dropped is sometimes called Welch's ttest. These tests are often referred to as "unpaired" or "independent samples" ttests, as they are typically applied when the statistical units underlying the two samples being compared are nonoverlapping.^{[5]}
 A test of the null hypothesis that the difference between two responses measured on the same statistical unit has a mean value of zero. For example, suppose we measure the size of a cancer patient's tumor before and after a treatment. If the treatment is effective, we expect the tumor size for many of the patients to be smaller following the treatment. This is often referred to as the "paired" or "repeated measures" ttest:^{[5]}^{[6]} see paired difference test.
 A test of whether the slope of a regression line differs significantly from 0.
Assumptions
Most ttest statistics have the form T = Z/s, where Z and s are functions of the data. Typically, Z is designed to be sensitive to the alternative hypothesis (i.e. its magnitude tends to be larger when the alternative hypothesis is true), whereas s is a scaling parameter that allows the distribution of T to be determined.
As an example, in the onesample ttest , where is the sample mean of the data, n is the sample size, and σ is the population standard deviation of the data; s in the onesample ttest is , where is the sample standard deviation.
The assumptions underlying a ttest are that
 Z follows a standard normal distribution under the null hypothesis
 ps^{2} follows a χ^{2} distribution with p degrees of freedom under the null hypothesis, where p is a positive constant
 Z and s are independent.
In a specific type of ttest, these conditions are consequences of the population being studied, and of the way in which the data are sampled. For example, in the ttest comparing the means of two independent samples, the following assumptions should be met:
 Each of the two populations being compared should follow a normal distribution. This can be tested using a normality test, such as the ShapiroWilk or Kolmogorov–Smirnov test, or it can be assessed graphically using a normal quantile plot.
 If using Student's original definition of the ttest, the two populations being compared should have the same variance (testable using Levene's test, Bartlett's test, or the Brown–Forsythe test; or assessable graphically using a normal quantile plot). If the sample sizes in the two groups being compared are roughly equal, Student's original ttest is highly robust to the presence of unequal variances.^{[7]} Welch's ttest is insensitive to equality of the variances regardless of whether the sample sizes are similar.
 The data used to carry out the test should be sampled independently from the two populations being compared. This is in general not testable from the data, but if the data are known to be dependently sampled (i.e. if they were sampled in clusters), then the classical ttests discussed here may give misleading results.
Unpaired and paired twosample ttests
Main article: Paired difference testTwosample ttests for a difference in mean can be either unpaired or paired. Paired ttests are a form of blocking, and have greater power than unpaired tests when the paired units are similar with respect to "noise factors" that are independent of membership in the two groups being compared. In a different context, paired ttests can be used to reduce the effects of confounding factors in an observational study.
The unpaired, or "independent samples" ttest is used when two separate sets of independent and identically distributed samples are obtained, one from each of the two populations being compared. For example, suppose we are evaluating the effect of a medical treatment, and we enroll 100 subjects into our study, then randomize 50 subjects to the treatment group and 50 subjects to the control group. In this case, we have two independent samples and would use the unpaired form of the ttest. The randomization is not essential here—if we contacted 100 people by phone and obtained each person's age and gender, and then used a twosample ttest to see whether the mean ages differ by gender, this would also be an independent samples ttest, even though the data are observational.
Dependent samples (or "paired") ttests typically consist of a sample of matched pairs of similar units, or one group of units that has been tested twice (a "repeated measures" ttest). A typical example of the repeated measures ttest would be where subjects are tested prior to a treatment, say for high blood pressure, and the same subjects are tested again after treatment with a bloodpressure lowering medication.
A dependent ttest based on a "matchedpairs sample" results from an unpaired sample that is subsequently used to form a paired sample, by using additional variables that were measured along with the variable of interest.^{[8]} The matching is carried out by identifying pairs of values consisting of one observation from each of the two samples, where the pair is similar in terms of other measured variables. This approach is often used in observational studies to reduce or eliminate the effects of confounding factors.
Calculations
Explicit expressions that can be used to carry out various ttests are given below. In each case, the formula for a test statistic that either exactly follows or closely approximates a tdistribution under the null hypothesis is given. Also, the appropriate degrees of freedom are given in each case. Each of these statistics can be used to carry out either a onetailed test or a twotailed test.
Once a t value is determined, a pvalue can be found using a table of values from Student's tdistribution. If the calculated pvalue is below the threshold chosen for statistical significance (usually the 0.10, the 0.05, or 0.01 level), then the null hypothesis is rejected in favor of the alternative hypothesis.
Onesample ttest
In testing the null hypothesis that the population mean is equal to a specified value μ_{0}, one uses the statistic
where is the sample mean, s is the sample standard deviation of the sample and n is the sample size. The degrees of freedom used in this test is n − 1.
Slope of a regression line
Suppose one is fitting the model
 Y_{i} = α + βx_{i} + ε_{i},
where x_{i}, i = 1, ..., n are known, α and β are unknown, and ε_{i} are independent identically normally distributed random errors with expected value 0 and unknown variance σ^{2}, and Y_{i}, i = 1, ..., n are observed. It is desired to test the null hypothesis that the slope β is equal to some specified value β_{0} (often taken to be 0, in which case the hypothesis is that x and y are unrelated).
Let
Then
has a tdistribution with n − 2 degrees of freedom if the null hypothesis is true. The standard error of the angular coefficient:
can be written in terms of the residuals. Let
Then t_{score} is given by:
Independent twosample ttest
Equal sample sizes, equal variance
This test is only used when both:
 the two sample sizes (that is, the number, n, of participants of each group) are equal;
 it can be assumed that the two distributions have the same variance.
Violations of these assumptions are discussed below.
The t statistic to test whether the means are different can be calculated as follows:
where
Here is the grand standard deviation (or pooled standard deviation), 1 = group one, 2 = group two. The denominator of t is the standard error of the difference between two means.
For significance testing, the degrees of freedom for this test is 2n − 2 where n is the number of participants in each group.
Unequal sample sizes, equal variance
This test is used only when it can be assumed that the two distributions have the same variance. (When this assumption is violated, see below.) The t statistic to test whether the means are different can be calculated as follows:
where
Note that the formulae above are generalizations of the case where both samples have equal sizes (substitute n for n_{1} and n_{2}).
is an estimator of the common standard deviation of the two samples: it is defined in this way so that its square is an unbiased estimator of the common variance whether or not the population means are the same. In these formulae, n = number of participants, 1 = group one, 2 = group two. n − 1 is the number of degrees of freedom for either group, and the total sample size minus two (that is, n_{1} + n_{2} − 2) is the total number of degrees of freedom, which is used in significance testing.
Unequal sample sizes, unequal variance
This test also known as Welch's ttest is used only when the two population variances are assumed to be different (the two sample sizes may or may not be equal) and hence must be estimated separately. The t statistic to test whether the population means are different can be calculated as follows:
where
Where s^{2} is the unbiased estimator of the variance of the two samples, n = number of participants, 1 = group one, 2 = group two. Note that in this case, is not a pooled variance. For use in significance testing, the distribution of the test statistic is approximated as being an ordinary Student's t distribution with the degrees of freedom calculated using
This is called the Welch–Satterthwaite equation. Note that the true distribution of the test statistic actually depends (slightly) on the two unknown variances: see Behrens–Fisher problem.
Dependent ttest for paired samples
This test is used when the samples are dependent; that is, when there is only one sample that has been tested twice (repeated measures) or when there are two samples that have been matched or "paired". This is an example of a paired difference test.
For this equation, the differences between all pairs must be calculated. The pairs are either one person's pretest and posttest scores or between pairs of persons matched into meaningful groups (for instance drawn from the same family or age group: see table). The average (X_{D}) and standard deviation (s_{D}) of those differences are used in the equation. The constant μ_{0} is nonzero if you want to test whether the average of the difference is significantly different from μ_{0}. The degree of freedom used is n − 1.
Example of repeated measures Number Name Test 1 Test 2 1 Mike 35% 67% 2 Melanie 50% 46% 3 Melissa 90% 86% 4 Mitchell 78% 91% Example of matched pairs Pair Name Age Test 1 John 35 250 1 Jane 36 340 2 Jimmy 22 460 2 Jessy 21 200 Worked examples
Let A_{1} denote a set obtained by taking 6 random samples out of a larger set:
and let A_{2} denote a second set obtained similarly:
These could be, for example, the weights of screws that were chosen out of a bucket.
We will carry out tests of the null hypothesis that the means of the populations from which the two samples were taken are equal.
The difference between the two sample means, each denoted by , which appears in the numerator for all the twosample testing approaches discussed above, is
The sample standard deviations for the two samples are approximately 0.05 and 0.11, respectively. For such small samples, a test of equality between the two population variances would not be very powerful. Since the sample sizes are equal, the two forms of the two sample ttest will perform similarly in this example.
Unequal variances
If the approach for unequal variances (discussed above) is followed, the results are
and
The test statistic is approximately 1.959. The twotailed test pvalue is approximately 0.091 and the onetailed pvalue is approximately 0.045.
Equal variances
If the approach for equal variances (discussed above) is followed, the results are
and
Since the sample sizes are equal (both are 6), the test statistic is again approximately equal to 1.959. Since the degrees of freedom is different from what it is in the unequal variances test, the pvalues will differ slightly from what was found above. Here, the twotailed pvalue is approximately 0.078, and the onetailed pvalue is approximately 0.039. Thus if there is good reason to believe that the population variances are equal, the results become somewhat more suggestive of a difference in the mean weights for the two populations of screws.
Alternatives to the ttest for location problems
The ttest provides an exact test for the equality of the means of two normal populations with unknown, but equal, variances. (The Welch's ttest is a nearlyexact test for the case where the data are normal but the variances may differ.) For moderately large samples and a one tailed test, the t is relatively robust to moderate violations of the normality assumption.^{[9]}
For exactness, the ttest and Ztest require normality of the sample means, and the ttest additionally requires that the sample variance follows a scaled χ^{2} distribution, and that the sample mean and sample variance be statistically independent. Normality of the individual data values is not required if these conditions are met. By the central limit theorem, sample means of moderately large samples are often wellapproximated by a normal distribution even if the data are not normally distributed. For nonnormal data, the distribution of the sample variance may deviate substantially from a χ^{2} distribution. However, if the sample size is large, Slutsky's theorem implies that the distribution of the sample variance has little effect on the distribution of the test statistic. If the data are substantially nonnormal and the sample size is small, the ttest can give misleading results. See Location test for Gaussian scale mixture distributions for some theory related to one particular family of nonnormal distributions.
When the normality assumption does not hold, a nonparametric alternative to the ttest can often have better statistical power. For example, for two independent samples when the data distributions are asymmetric (that is, the distributions are skewed) or the distributions have large tails, then the Wilcoxon Rank Sum test (also known as the MannWhitney U test) can have three to four times higher power than the ttest.^{[9]}^{[10]}^{[11]} The nonparametric counterpart to the paired samples t test is the Wilcoxon signedrank test for paired samples. For a discussion on choosing between the t and nonparametric alternatives, see Sawilowsky.^{[12]}
Oneway analysis of variance generalizes the twosample ttest when the data belong to more than two groups.
Multivariate testing
A generalization of Student's t statistic, called Hotelling's Tsquare statistic, allows for the testing of hypotheses on multiple (often correlated) measures within the same sample. For instance, a researcher might submit a number of subjects to a personality test consisting of multiple personality scales (e.g. the Big Five). Because measures of this type are usually highly correlated, it is not advisable to conduct separate univariate ttests to test hypotheses, as these would neglect the covariance among measures and inflate the chance of falsely rejecting at least one hypothesis (Type I error). In this case a single multivariate test is preferable for hypothesis testing. Hotelling's T^{ 2} statistic follows a T^{ 2} distribution. However, in practice the distribution is rarely used, and instead converted to an F distribution.
Onesample T^{ 2} test
For a onesample multivariate test, the hypothesis is that the mean vector () is equal to a given vector (). The test statistic is defined as:
where n is the sample size, is the vector of column means and is a sample covariance matrix.
Twosample T^{ 2} test
For a twosample multivariate test, the hypothesis is that the mean vectors (, ) of two samples are equal. The test statistic is defined as
Implementations
Most spreadsheet programs and statistics packages, such as QtiPlot, OpenOffice.org Calc, Microsoft Excel, SAS, SPSS, Stata, DAP, gretl, R, Python ([1]), PSPP, and Minitab, include implementations of Student's ttest.
See also
 Student's tstatistic
 Ftest
 Conditional change model
Further reading
 Boneau, C. Alan (1960). "The effects of violations of assumptions underlying the t test". Psychological Bulletin 57 (1): 49–64. doi:10.1037/h0041412
 Edgell, Stephen E., & Noon, Sheila M (1984). "Effect of violation of normality on the t test of the correlation coefficient". Psychological Bulletin 95 (3): 576–583. doi:10.1037/00332909.95.3.576.
Notes
 ^ Richard Mankiewicz, The Story of Mathematics (Princeton University Press), p.158.
 ^ ^{a} ^{b} O'Connor, John J.; Robertson, Edmund F., "Student's ttest", MacTutor History of Mathematics archive, University of St Andrews, http://wwwhistory.mcs.standrews.ac.uk/Biographies/Gosset.html.
 ^ Fisher Box, Joan (1987). "Guinness, Gosset, Fisher, and Small Samples". Statistical Science 2 (1): 45–52. doi:10.1214/ss/1177013437. JSTOR 2245613.
 ^ Raju TN (2005). "William Sealy Gosset and William A. Silverman: two "students" of science". Pediatrics 116 (3): 732–5. doi:10.1542/peds.20051134. PMID 16140715.
 ^ ^{a} ^{b} Fadem, Barbara (2008). HighYield Behavioral Science (HighYield Series). Hagerstwon, MD: Lippincott Williams & Wilkins. ISBN 0781782589.
 ^ Zimmerman, Donald W. (1997). "A Note on Interpretation of the PairedSamples t Test". Journal of Educational and Behavioral Statistics 22 (3): 349–360. JSTOR 1165289.
 ^ Markowski, Carol A; Markowski, Edward P. (1990). "Conditions for the Effectiveness of a Preliminary Test of Variance". The American Statistician 44 (4): 322–326. doi:10.2307/2684360. JSTOR 2684360.
 ^ David, HA; Gunnink, Jason L (1997). "The Paired t Test Under Artificial Pairing". The American Statistician 51 (1): 9–12. doi:10.2307/2684684. JSTOR 2684684.
 ^ ^{a} ^{b} Sawilowsky S., Blair R. C. (1992). "A more realistic look at the robustness and type II error properties of the t test to departures from population normality". Psychological Bulletin 111 (2): 353–360. doi:10.1037/00332909.111.2.352.
 ^ Blair, R. C.; Higgins, J.J. (1980). "A comparison of the power of Wilcoxon’s ranksum statistic to that of Student’s t statistic under various nonnormal distributions.". Journal of Educational Statistics 5 (4): 309–334. doi:10.2307/1164905. JSTOR 1164905.
 ^ Fay, MP; Proschan, MA (2010). "WilcoxonMannWhitney or ttest? On assumptions for hypothesis tests and multiple interpretations of decision rules". Statistics Surveys 4: 1–39. doi:10.1214/09SS051. PMC 2857732. PMID 20414472. http://www.ijournals.org/ss/viewarticle.php?id=51.
 ^ Sawilowsky S (2005). "Misconceptions leading to choosing the t test over the Wilcoxon MannWhitney U test for shift in location parameter". Journal of Modern Applied Statistical Methods 4 (2): 598–600.
References
 O'Mahony, Michael (1986). Sensory Evaluation of Food: Statistical Methods and Procedures. CRC Press. pp. 487. ISBN 0824773373.
 Press, William H.; Saul A. Teukolsky, William T. Vetterling, Brian P. Flannery (1992). Numerical Recipes in C: The Art of Scientific Computing. Cambridge University Press. pp. p. 616. ISBN 0521431085. http://www.nr.com/.
External links
Online calculators
 Online TTest Calculator
 2 Sample TTest Calculator
 GraphPad's Paired/Unpaired/Welch TTest Calculator
Categories: Statistical tests
 Statistical methods
 Parametric statistics
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