All statistics and graphs for Comparisons

N

The sample size (N) is the total number of observations in each group.

Interpretation

The sample size affects the confidence interval and the power of the test.

Usually, a larger sample yields a narrower confidence interval. A larger sample size also gives the test more power to detect a difference.

Mean

The mean of the observations within each group. The mean describes each group with a single value identifying the center of the data. It is the sum of all the observations with a group divided by the number of observations in that group.

Interpretation

The mean of each sample provides an estimate of each population mean. The differences between sample means are the estimates of the difference between the population means.

Because the difference between the group means are based on data from a sample and not the entire population, you cannot be certain it equals the population difference. To get a better sense of the population difference, you can use the confidence interval.

Grouping

Use the Grouping Information table to quickly determine whether the mean difference between any pair of groups is statistically significant.

The Grouping column contains letters that group the factor levels. Groups that do not share a letter have a mean difference that is statistically significant.

If the table identifies differences that are statistically significant, use the confidence intervals of the differences to determine whether the differences are practically significant.

You can do a multiple comparison analysis for random terms to determine which levels of the term are significantly different from the other levels. For example, if you study the effectiveness of a drug on a particular illness, subject is usually a significant random factor. You can use multiple comparisons to determine whether the drug affected specific subjects in the study differently (perhaps it made one of them more ill).

Interpretation

Grouping Information Using the Tukey Method and 95% Confidence Paint N Mean Grouping Blend 4 6 18.07 A Blend 1 6 14.73 A B Blend 3 6 12.98 A B Blend 2 6 8.57 B Means that do not share a letter are significantly different.

In these results, the table shows that group A contains Blends 1, 3, and 4, and group B contains Blends 1, 2, and 3. Blends 1 and 3 are in both groups. Differences between means that share a letter are not statistically significant. Blends 2 and 4 do not share a letter, which indicates that Blend 4 has a significantly higher mean than Blend 2.

Fisher Individual Tests for Differences of Means

Use the individual confidence intervals to identify statistically significant differences between the group means, to determine likely ranges for the differences, and to determine whether the differences are practically significant. Fisher's individual tests table displays a set of confidence intervals for the difference between pairs of means.

The individual confidence level is the percentage of times that a single confidence interval includes the true difference between one pair of group means, if you repeat the study. Individual confidence intervals are available only for Fisher's method. All of the other comparison methods produce simultaneous confidence intervals.

Controlling the individual confidence level is uncommon because it does not control the simultaneous confidence level, which often increases to unacceptable levels. If you do not control the simultaneous confidence level, the chance that at least one confidence interval does not contain the true difference increases with the number of comparisons.

The confidence interval of the difference is composed of the following two parts:

Point estimate
The point estimate is the difference between a pair of means and is calculated from the sample data. The confidence interval is centered on this value.
Error margin
The error margin defines the width of the confidence interval and is determined by the observed variability in the sample and the confidence level. To calculate the upper limit of the confidence interval, the error margin is added to the point estimate. To calculate the lower limit of the confidence interval, the error margin is subtracted from the point estimate.

Interpretation

Use the confidence intervals to assess the differences between group means.

Fisher Individual Tests for Differences of Means Difference SE of Difference of Levels of Means Difference 95% CI T-Value Blend 2 - Blend 1 -6.17 2.28 (-10.92, -1.41) -2.70 Blend 3 - Blend 1 -1.75 2.28 ( -6.51, 3.01) -0.77 Blend 4 - Blend 1 3.33 2.28 ( -1.42, 8.09) 1.46 Blend 3 - Blend 2 4.42 2.28 ( -0.34, 9.17) 1.94 Blend 4 - Blend 2 9.50 2.28 ( 4.74, 14.26) 4.17 Blend 4 - Blend 3 5.08 2.28 ( 0.33, 9.84) 2.23 Adjusted Difference of Levels P-Value Blend 2 - Blend 1 0.014 Blend 3 - Blend 1 0.452 Blend 4 - Blend 1 0.159 Blend 3 - Blend 2 0.067 Blend 4 - Blend 2 0.000 Blend 4 - Blend 3 0.037 Simultaneous confidence level = 80.83%

The confidence intervals indicate the following:
  • The confidence interval for the difference between the means of Blend 4 and 2 extends from 4.74 to 14.26. This range does not include zero, which indicates that the difference between these means is statistically significant.
  • The confidence interval for the difference between the means of Blend 2 and 1 extends from -10.92 to -1.41. This range does not include zero, which indicates that the difference between these means is statistically significant.
  • The confidence interval for the difference between the means of Blend 4 and 3 extends from 0.33 to 9.84. This range does not include zero, which indicates that the difference between these means is statistically significant.
  • The confidence intervals for all the remaining pairs of means include zero, which indicates that the differences are not statistically significant.
  • The 95% individual confidence level indicates that you can be 95% confident that each confidence interval contains the true difference for that specific comparison. However, the simultaneous confidence level indicates that you can be only 80.83% confident that all the intervals contain the true differences.

Difference of Means

This value is the difference between the sample means of two groups.

Interpretation

The differences between the sample means of the groups are estimates of the differences between the populations of these groups.

Because each mean difference is based on data from a sample and not from the entire population, you cannot be certain that it equals the population difference. To better understand the differences between population means, use the confidence intervals.

SE of Difference

The standard error of the difference between means (SE of Difference) estimates the variability of the difference between sample means that you would obtain if you took repeated samples from the same populations.

Interpretation

Use the standard error of the difference between means to determine how precisely the differences between the sample means estimate the differences between the population means. A lower standard error value indicates a more precise estimate.

Minitab uses the standard error of the difference to calculate the confidence intervals of the differences between means, which is a range of values that is likely to include the population differences.

DF

The degrees of freedom (DF) are the amount of information in your data. Minitab uses the degrees of freedom to calculate the t-test for the difference in means. Minitab only displays the degrees of freedom if you perform comparisons for a mixed effects model.

Simultaneous 95% CI

Use the simultaneous confidence intervals of the difference (95% CI) to identify mean differences that are statistically significant, to determine likely ranges for the differences, and to assess the practical significance of the differences. The table displays a set of confidence intervals for the difference between pairs of means. Confidence intervals that do not contain zero indicate a mean difference that is statistically significant.

The simultaneous confidence level is the percentage of times that a set of confidence intervals includes the true differences for all group comparisons if the study were repeated multiple times.

Controlling the simultaneous confidence level is particularly important when you perform multiple comparisons. If you do not control the simultaneous confidence level, the chance that at least one confidence interval does not contain the true difference increases with the number of comparisons.

For more information, go to Understanding individual and simultaneous confidence levels in multiple comparisons.

The confidence interval of the difference is composed of the following two parts:

Point estimate
The point estimate is the difference between a pair of means and is calculated from the sample data. The confidence interval is centered on this value.
Margin of error
The margin of error defines the width of the confidence interval and is determined by the observed variability in the sample, the sample size, and the confidence level. To calculate the upper limit of the confidence interval, the error margin is added to the point estimate. To calculate the lower limit of the confidence interval, the error margin is subtracted from the point estimate. The margin widens as the number of comparisons increases in order to maintain the simultaneous level of confidence that all intervals contain the true population differences.

You can do a multiple comparison analysis for random terms to determine which levels of the term are significantly different from the other levels. For example, if you study the effectiveness of a drug on a particular illness, subject is usually a significant random factor. You can use multiple comparisons to determine whether the drug affected specific subjects in the study differently (perhaps it made one of them more ill).

Interpretation

Use the confidence intervals to assess the differences between group means.

Tukey Simultaneous Tests for Differences of Means Difference of Subject Difference SE of Simultaneous 95% Adjusted Levels of Means Difference CI T-Value P-Value 2 - 1 0.6567 0.0664 ( 0.4768, 0.8366) 9.89 0.000 3 - 1 0.9633 0.0708 ( 0.7715, 1.1551) 13.60 0.000 4 - 1 0.6900 0.0750 ( 0.4870, 0.8930) 9.20 0.000 3 - 2 0.3067 0.0632 ( 0.1356, 0.4777) 4.86 0.000 4 - 2 0.0333 0.0678 (-0.1502, 0.2169) 0.49 0.960 4 - 3 -0.2733 0.0721 (-0.4686, -0.0781) -3.79 0.003 Individual confidence level = 98.94%

The simultaneous confidence intervals indicate the following:
  • The confidence interval for the difference between the means of Subject 4 and 2 is -0.1502 to 0.2169. This range includes zero, which indicates that the difference is not statistically significant.
  • The confidence intervals for the remaining pairs of means do not include zero, which indicates that these differences are statistically significant.
  • The 95% simultaneous confidence level indicates that you can be 95% confident that all the confidence intervals contain the true differences.
  • The table indicates that the individual confidence level is 98.94%. This result indicates that you can be 98.94% confident that each individual interval contains the true difference between a specific pair of group means.
  • The 98.94% individual confidence level for each comparison collectively produces the 95% simultaneous confidence level for all six comparisons.

T-value

The t-value is a test statistic that measures the ratio between the difference in means and the standard error of the difference.

Interpretation

You can use the t-value to determine whether to reject the null hypothesis, which states that the difference in means is 0. However, most people use the p-value because it is easier to interpret. For more information on using the critical value, go to Using the t-value to determine whether to reject the null hypothesis.

Minitab uses the t-value to calculate the p-value.

Adjusted p-value

The adjusted p-value indicates which pairs within a family of comparisons are significantly different. The adjustment limits the family error rate to the alpha level that you specify. If you use a regular p-value for multiple comparisons, the family error rate increases with each additional comparison.

It is important to consider the family error rate when making multiple comparisons, because your chances of committing a type I error for a series of comparisons is greater than the error rate for any one comparison alone.

Interpretation

If the adjusted p-value is less than alpha, reject the null hypothesis and conclude that the difference between a pair of group means is statistically significant. The adjusted p-value also represents the smallest family error rate at which a particular null hypothesis is rejected.

Interval plot for differences of means

Use the confidence intervals to determine likely ranges for the differences and to assess the practical significance of the differences. The graph displays a set of confidence intervals for the difference between pairs of means. Confidence intervals that do not contain zero indicate a mean difference that is statistically significant.

Depending on the comparison method you chose, the plot compares different pairs of groups and displays one of the following types of confidence intervals.

  • Individual confidence level

    The percentage of times that a single confidence interval would include the true difference between one pair of group means if the study were repeated multiple times.

  • Simultaneous confidence level

    The percentage of times that a set of confidence intervals would include the true differences for all group comparisons if the study were repeated multiple times.

    Controlling the simultaneous confidence level is particularly important when you perform multiple comparisons. If you do not control the simultaneous confidence level, the chance that at least one confidence interval does not contain the true difference increases with the number of comparisons.

For more information, go to Understanding individual and simultaneous confidence levels in multiple comparisons.

You can do a multiple comparison analysis for random terms to determine which levels of the term are significantly different from the other levels. For example, if you study the effectiveness of a drug on a particular illness, subject is usually a significant random factor. You can use multiple comparisons to determine whether the drug affected specific subjects in the study differently (perhaps it made one of them more ill).

Interpretation

Use the confidence intervals to assess the differences between group means.

Tukey Simultaneous Tests for Differences of Means Difference of Subject Difference SE of Simultaneous 95% Adjusted Levels of Means Difference CI T-Value P-Value 2 - 1 0.6567 0.0664 ( 0.4768, 0.8366) 9.89 0.000 3 - 1 0.9633 0.0708 ( 0.7715, 1.1551) 13.60 0.000 4 - 1 0.6900 0.0750 ( 0.4870, 0.8930) 9.20 0.000 3 - 2 0.3067 0.0632 ( 0.1356, 0.4777) 4.86 0.000 4 - 2 0.0333 0.0678 (-0.1502, 0.2169) 0.49 0.960 4 - 3 -0.2733 0.0721 (-0.4686, -0.0781) -3.79 0.003 Individual confidence level = 98.94%

The simultaneous confidence intervals indicate the following:
  • The confidence interval for the difference between the means of Subject 4 and 2 is -0.1502 to 0.2169. This range includes zero, which indicates that the difference is not statistically significant.
  • The confidence intervals for the remaining pairs of means do not include zero, which indicates that these differences are statistically significant.
  • The 95% simultaneous confidence level indicates that you can be 95% confident that all the confidence intervals contain the true differences.
  • The table indicates that the individual confidence level is 98.94%. This result indicates that you can be 98.94% confident that each individual interval contains the true difference between a specific pair of group means.
  • The 98.94% individual confidence level for each comparison collectively produces the 95% simultaneous confidence level for all six comparisons.

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