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As is widely known, ANOVA (Analysis of Variance) is the most representative method of evaluating the significance and impact of categorical factors for continuous response. is. Types of ANOVA include OneWay ANOVA and Two-Way ANOVA. In this case, one-way or two-way means the number of factors. For example, if you test the difference in yields according to materials A and B, the factor is one-way analysis of variance because it is one material, whereas if you want to test the difference in yields at 100 and 110 degrees of heating temperature of materials A and B, the factor is material And temperature, so this case is called a two-way analysis of variance. The most basic form of ANOVA can be seen when there are two levels of each factor, such as materials A and B or temperature 10 degrees and 110 degrees. In this case, multiple comparisons when there are three or more levels of the factor. Let's take a look.
Generally Analysis of Variance (ANOVA) The hypothesis is as follows when testing the difference between means using.
1) Null hypothesis (Ho): The average of all groups is the same.
2) Alternative hypothesis (H1): The mean of all groups is not the same.
The key is to reject the null hypothesis if at least one group's mean is different from the other's mean. In other words, rejecting the null hypothesis does not mean that there is a difference in the mean of all groups. like this Multiple Comparison Let's find out what problems can arise in case.
For example, assuming that an X factor with five groups of A, B, C, D, and E is tested for the difference in mean value between two groups using the ANOVA method, 10 comparisons are possible in all. At this time, the risk rate for the entire test is 0.401 based on the significance level (α, type 1 error) 0.05. Because when the significance level (type 1 error) is 0.05, the probability that the type 1 error does not occur is (1-0.05) = 0.95, and the probability that the type 1 error does not occur for 10 comparisons is (0.95) ) 10 = 0.599.
In other words, the probability of occurrence of a type 1 error in at least one comparison in 10 comparisons is 1 – 0.599 = 0.401, which increases the likelihood of interpreting that there is a significant difference even though it is not significant. In the case of multiple comparisons, a general ANOVA It needs a different method and interpretation than in the case of.
Let's take a look at the sample data below in JMP.
(Help / Sample Data Library / big class.jmp)
To see if the average height varies with age Analyze / Fit Y by X Select age as X and height as Y and click OK. ▼ Oneway ~ / Means / Anova If you run, the result is as below.
In the above result, you should not judge that the height difference depends on the age by looking only at the P Value (0.0008). This means that there is a difference between at least one of the six age levels (12 to 17 years old), not all age levels, so be careful.
In the case of three or more multiple comparisons like this, ▼ Oneway ~ / Compare Means You should use one of the five methods below. Let's do all five methods here. (You can do all five methods at once by holding down the alt Key and clicking the red triangle next to Oneway.)
The five methods are briefly summarized below.
One. Each Pairs, Student's t
One) LSD Threshold Matrix Means that there is a significant difference between levels expressed as positive values.
2) Connecting Letters Report In, it means that there is no significant difference between Levels containing the same character.
3) Ordered Difference Report Is a summary of the test results between the two levels.
4) ▼ Comparisons / Detailed Comparison Report If you click, you can check all the comparison results between the two levels (here, there are 6 levels to be compared, so all 15 comparison results).
2. All Pairs, Tukey HSD (Honestly Significant Difference)
Compared to the Student's t test above, the results are quite conservative. (Since statistical significance is interpreted very strictly, it is widely used in practical situations where such interpretation is required.)
3. With Best, HSu MCB
This method compares to the Group with the minimum or maximum sample size, so it shows slightly different test results than the two methods above.
4. With Control, Dunnett's
This method is a test based on one comparison group (here 12 years old is selected as the comparison group).
5. Each Pair Stepwise Newman Keuls(SNK) The result of is as follows.
6. Comprehensive interpretation
1) In practice Student's T method and Tukey-Kramer (Tukey HSD) method are most used Seems to be.
2) Among these Tukey-Kramer method is more conservative and strict Do it.
3) If you click any one of the circles at the top, levels without significant difference are displayed in the same red color, and levels with significant difference are displayed in black. Below is the result of choosing the 13 year old of TurkeyKramer method. (The last method, Stepwise Newman Keuls (SNK), does not display these Circles.)
