T-Test

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REQUIREMENTS: The T-statistic is used to test differences in the means of two groups. The grouping variable is categorical and data for the dependent variable is interval scaled. The following table shows alternative statistical techniques that can be used to analyze 2 groups.

Contents

[edit] Selected Nonparametric and Parametric Statistical Tests for Two-Sample Cases

Image:ttimg035.jpg

The t-distribution was developed by W. S. Gosset (1908)[1]. As an employee of the Guinness brewery in Dublin, Gosset was not permitted to publish research findings in his own name, and hence penned the pseudonym "Student". The t-distribution, as it was first designated, has been known under a variety of names, including the Student's distribution and Student's t-distribution.

The t-distribution revolutionized statistics and the ability to work with small samples. Prior to this time, statistical work was based largely on the value of z, which was used to designate a point on the normal distribution where population parameters were known. The z value is the deviation of the sample mean from the mean of the population and is expressed in terms of variance within a normally distributed population.[2]

Image:ttimg036.gif

The purpose of the z value is to express the amount of deviation between the sample mean and the population mean and to permit the making of inferences as to whether the sample mean belongs to the population in question.

The characteristics of the population (μ and σ), to which we desire to make inferences, are rarely known. This state of perfect knowledge assumed by the z test makes the use of this statistic difficult to justify.

The t statistic does not provide the population variance information needed for the z test, but instead uses the sample variance (and sample standard deviation).

Image:ttimg037.gif

The t-distribution is symmetrical about the mean and is approximately normal. It is centered at the population mean of 0 and for large samples has a variance σ = 1.

The central limit theorem tells us that the sampling distribution of all possible sample means (bar), approaches normality as the size of the samples increase. This is true, even when the population is not normally distributed. The t-distribution rapidly approaches the shape of a normal distribution as sample size increases. The t-distribution is considered normal when n=30.

The t-distribution is used to make inferences concerning the difference between the two populations μ1 and μ2. The specific statistical theory used relates to the distribution of differences between the two sets of independent sample means, the sampling distribution of Image:ttimg038.gif

[edit] MATHEMATICAL COMPUTATIONS FOR THE T-TEST

  1. Statistical analysis programs compute t-statistics and associated probability levels for the equality of the means of two groups based on pooled and separate variance estimates. An F-statistic and associated probability level for the equality of group variances is also computed. Groups may be defined by specifying codes to be included. Several dependent variables may often be analyzed concurrently. Paired comparison t-ratios may be obtained through the use of IF and RECODE commands (SPSS).
  2. Output from this program includes:
    1. F - ratio of variance
    2. t - value (based on pooled variance estimate)
    3. t - value (based on separate variance estimate)
    4. Two-tailed probability levels for each t and for the F
    5. Means
    6. Standard deviations
    7. Standard error of the means
    8. Number of observations included in computing 5-7 above
    9. Optional output of data input
      1. Pooling is required in each of the following two cases:
        1. When we are testing for the same population proportion in two populations.
        2. When testing the difference in means between two small samples.
          In case of pooling, we pool the point estimate by simple averages. The pooled standard error is given by:
          Image:ttimg039.gif

[edit] COMPUTATIONAL PROCEDURE

P variables for the first case are read. In the data manipulation, if Xi or Xj is missing, then Xk is missing. If a case meets the specifications for a specific t-test analysis, the observation will be included in that calculation. Each problem is divided into two groups: an X and a Y category. For each analysis, the number of non-missing observations, the mean, standard deviation, and standard error are computed for each variable of each category. The t-values, F-values, and corresponding probability level for between-category comparison are computed for each variable.

[edit] Step 1. 
Xijk,i = 1,2,...,n
     j = 1,2,...,(p+q)
     k = 1,2; 1 = X category, 2 = Y category
[edit] Step 2.

Image:ttimg040.gif
Mean
Image:ttimg041.gif
Standard deviation Image:ttimg042.gif
Standard error (of Mean) Image:ttimg043.gif
Image:ttimg044.gif
Image:ttimg045.gif

Degrees of freedom based on pooled variance estimate: Image:ttimg046.gif.

t based on pooled variance estimate:
Image:ttimg047.jpg

t based on separate variance estimate:
Image:ttimg048.jpg

Degrees of freedom based on separate variance estimate:
Image:ttimg049.gif 

DEPENDENT VARIABLE IS : V64      EDUCATION SPOUSE                        

GROUP 1 IS LESS THAN OR EQUAL TO     2 AND GROUP 2 IS GREATER THAN     2

                                                         * POOLED VARIANCE ESTIMATE1 *   SEPARATE VARIANCE ESTIMATE

VAR.    NO.OF          STD.      STD.   *   F2      P3   *  T      DEGREES       P   *   T     DEGREES       P 

INDEX   CASES4 MEAN    DEVN.    ERROR   * VALUE    VALUE * VALUE   OF FREEDOM   VALUE*  VALUE  OF FREEDOM  VALUE  *********************************************************************************************************************

 1   V18   COMPETENCE OF NURSING STAFF  *                *

CODE 1    68   1.0147  .121      .015   *                *                           *

                                        * 1.69     .014  *  .38      181       .707  *   .35     113.86      .726

CODE 2    115  1.0087  .093      .009   *                *                           *

 ---------------------------------------------------------------------------------------------------------------------

 2   V19      PROMPTNESS OF RESPONSE    *                *

CODE 1    67   1.0299  .171      .021   *                *                           *

                                        *   3.29   .000  * 1.05      177       .294  *    .92    90.39       .361

CODE 2    112  1.0089  .094      .009   *                *                           *

--------------------------------------------------------------------------------------------------------------------- 

 3   V20      FRIENDLINESS OF STAFF     *                *

CODE 1    68   1.0147  .121      .015   *                *                           *

                                        *   1.17   .452  * -.14      181       .891  *  -.14     149.78      .889

CODE 2   115   1.0174  .131      .012   *                *                           *

---------------------------------------------------------------------------------------------------------------------

 4   V21      NURSE EXPLAINS TO PATIENT *                *

CODE 1    65   1.0615   242      .030   *                *                           *

                                        *   1.72   .012  *  .82       177      .413  *  .76      106.88      .447

CODE 2   114   1.0351  .185      .017   *                *                           *

---------------------------------------------------------------------------------------------------------------------

 5   V22      DR EXPLAINS TO PATIENT    *                *

CODE 1    67   1.0149  .122      .015   *                *                           *

                                        *    .00  1.000  *  1.32       181     .189  *  1.00       66.00      .321

CODE 2    116  1.0000  .000      .000   *                *                           *


  1. In case of pooling, the pooled point estimate and pooled standard error are calculated. In case of the separate variance estimate, the difference between the two groups is used as the data. The mean and standard error of the difference are computed.
  2. F-value is the ratio of the mean square error (mean of the sum of squares) for each of the two groups.
  3. The P-value is the significance level corresponding to the computed F-value. If this P-value is less than or equal to the significance level of the test, then we reject the null hypothesis.
  4. The two groups are defined by the cut-off value selected. The cases are assigned to the two groups as per the cut-off value.

[edit] Footnotes:

  1. ↑ Student (1908), "The Probable Error of a Mean", Biometrica, 6:1.
  2. ↑ The normal probability distribution is defined by the equation: Image:ttimg050.gif and Image:ttimg051.gif
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