學生t檢驗
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學生t檢驗是指虛無假說成立時的任一檢定統計有學生t-分佈的統計假說檢定。
目录 |
[编辑] 由來
t檢驗是戈斯特為了觀測釀酒品質而發明的，「學生」則是他的筆名。戈斯特在位於都柏林的健力士釀酒廠擔任統計學家，基於Claude Guinness聘用從牛津大學和劍橋大學出來的最好的畢業生以將生物化學及統計學應用到健力士工業程序的創新政策。戈特特於1908年在Biometrika上公布t檢驗，但因其老闆認為其為商業機密而被迫使用筆名。實際上，戈斯特的真實身份不只是其他統計學家不知道，連其老闆也不知道－the company insisted on the pseudonym so that it could turn a blind eye to the breach of its rules.
今日，它更常被應用在能由小樣本判斷的信心。
[编辑] 應用
Among the most frequently used t tests are:
- A test of the null hypothesis that the means of two normally distributed populations are equal. Given two data sets, each characterized by its mean, standard deviation and number of data points, we can use some kind of t test to determine whether the means are distinct, provided that the underlying distributions can be assumed to be normal. All such tests are usually called Student's t tests, 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 t test. There are different versions of the t test depending on whether the two samples are
- independent of each other (e.g., individuals randomly assigned into two groups), or
- paired, so that each member of one sample has a unique relationship with a particular member of the other sample (e.g., the same people measured before and after an intervention, or IQ test scores of a husband and wife).
- If the t value that is calculated is above the threshold chosen for statistical significance (usually the 0.05 level), then the null hypothesis that the two groups do not differ is rejected in favor of an alternative hypothesis, which typically states that the groups do differ.
- A test of whether the mean of a normally distributed population has a value specified in a null hypothesis.
- A test of whether the slope of a regression line differs significantly from 0.
[编辑] 假设条件
- normal distribution of data (e.g. Wilk-Shapiro normality test)
- equality of variances (F test, or more robust Levene's test)
- Samples may be independent or dependent, depending on the hypothesis and the type of samples:
- Independent samples are usually two, randomly selected groups
- Dependent samples are either two groups matched on some variable (for example, age) or are the same people being tested twice (called repeated measures)
[编辑] 计算
[编辑] 独立t檢驗
[编辑] 样本大小不同时
[编辑] 样本大小一致时
[编辑] 相关t檢驗
[编辑] 怎样决定使用哪一个t檢驗
Whether the data points are normally distributed can be assessed by a normality test, such as Kolmogorov-Smirnov or Shapiro-Wilk.
Whether the sample variances are equal can be assessed using Bartlett's test, Levene's test, or the Brown & Forsythe test. However, it is probably statistically conservative not to make this assumption: modern statistical packages make the test equally easy to do with or without it. (Since all calculations are done subject to the null hypothesis, it may be very difficult to come up with a reasonable null hypothesis that accounts for equal means in the presence of unequal variances. In the usual case, the null hypothesis is that the different treatments have no effect; this makes unequal variances untenable. In this case, one should forgo the ease of using this variant afforded by the statistical packages. See also Behrens-Fisher problem.)
For novices, the most difficult issue is often whether the samples are paired (dependent) or independent. Dependent samples are sometimes described as involving "repeated measures", often arising when we make before and after measurements on the same individuals or objects. But related samples also occur in other cases. For example, to compare the heights of men and women, we might recruit 100 married couples, and compare the height of each woman with her partner; this would call for a related samples test. There are repeated measures here: the couple is measured twice - once for the woman and once for the man. Alternatively, we might recruit 100 men and 100 women, with no relationship between any particular man and any particular woman; in this case we would use an independent samples test.
Paired t test is also used for matched samples, where we are comparing two groups and performed matching so that for every person in group 1 there is a matched person in group 2.
[编辑] t檢驗之外的其它选择
If a non-parametric alternative to the t test is wanted, the usual choices are:
- for independent samples, the Mann-Whitney U test
- for related samples, either the binomial test or the Wilcoxon signed-rank test
