Running Fisher’s LSD Multiple Comparison Test in Excel For finding Inter-Groups Differences after getting significant results in overall ANOVA test

Fisher’s LSD Test It is also a popular Post Hoc Test. (Post Hoc = unplanned before experiment) It is the most ‘sensitive’ Post Hoc Test and most unlikely to miss sign. mean group diff. But the chance of committing Type I error - It can be used in either Equal or Unequal Sample size conditions

Part One One-Way Anova

e.g. Completely Randomized Design Unequal Sample Size An experiment with Completely Randomized Design has been started with 10 equal weight chickens in each group of Treatment A (Control), Treatment B, Treatment C and Treatment D, with increasing dosage of a new drug that might increase growing rate. However, some chickens have died during the experiment, especially in groups with higher dosage. Please find any significant different increase of weight among the 4 groups.

In Excel with ‘Analysis ToolPak’ Add-In activated, click Data, Data Analysis :-

Choose ‘Anova: Single Factor’ = One-way Anova

Select Data Area including Labels:-

Overall Anova result:- The overall Anova result reject the null hypothesis that all group means are equal! For finding exactly where the differences exist, we proceed to run Scheffe’s Test!! N Group Means to be used later Within Degree of Freedom DFw Within Group Variance MSw to be used later

Calculation of LSD (Lease Significant Difference) For Group 1 vs Group 2 LSD = t0.05/2,DFW 𝑀𝑆𝑊 1 𝑁1 + 1 𝑁2 = t0.025, 24 1.2408 1 10 + 1 8 = 2.064 X 0.5280 = 1.0749 I Mean Group 1 – Mean Group 2 I = 0.3009 < 1.0749 No significance is found on Mean Difference between Group 1 & Group 2!!

Calculation of LSD (Lease Significant Difference) For Group 1 vs Group 3 LSD = t0.05/2,DFW 𝑀𝑆𝑊 1 𝑁1 + 1 𝑁3 = t0.025, 24 1.2408 ( 1 10 + 1 6 ) = 2.064 X 0.5753 = 1.1874 I Mean Group 1 – Mean Group 3 I = 2.04907 > 1.1874 Significance is found on Mean difference between Group 1 & Group 3!!

Calculation of LSD (Lease Significant Difference) For Group 1 vs Group 4 LSD = t0.05/2,DFW 𝑀𝑆𝑊 1 𝑁1 + 1 𝑁4 = t0.025, 24 1.2408 ( 1 10 + 1 4 ) = 2.064 X 0.6590 = 1.3602 I Mean Group 1 – Mean Group 4 I = 1.8324 > 1.3602 Significance is found on Mean difference between Group 1 & Group 4!!

Calculation of LSD (Lease Significant Difference) For Group 2 vs Group 3 LSD = t0.05/2,DFW 𝑀𝑆𝑊 1 𝑁2 + 1 𝑁3 = t0.025, 24 1.2408 ( 1 8 + 1 6 ) = 2.064 X 0.6016 = 1.2417 I Mean Group 2 – Mean Group 3 I = 1.7482 > 1.2417 Significance is found on Mean difference between Group 2 & Group 3!!

Calculation of LSD (Lease Significant Difference) For Group 2 vs Group 4 LSD = t0.05/2,DFW 𝑀𝑆𝑊 1 𝑁2 + 1 𝑁4 = t0.025, 24 1.2408 ( 1 8 + 1 4 ) = 2.064 X 0.6821 = 1.4078 I Mean Group 2 – Mean Group 4 I = 1.5315 >1.4078 Significance is found on Mean difference between Group 2 & Group 4!!

Calculation of LSD (Lease Significant Difference) For Group 3 vs Group 4 LSD = t0.05/2,DFW 𝑀𝑆𝑊 1 𝑁3 + 1 𝑁4 = t0.025, 24 1.2408 ( 1 6 + 1 4 ) = 2.064 X 0.8007 = 1.6526 I Mean Group 3 – Mean Group 4 I = 0.2167 < 1.6526 No Significance is found on Mean difference between Group 3 & Group 4!!

Counter Checking with SPSS Using the same Data Set

Choose Post Hoc test e.g. LSD :-

Overall Anova result similiar to that in Excel:-

95% confidence Interval Lower Bound Upper Bound

Proving that the Excel result are exactly equal to that in SPSS!! Although the Excel result for the Fisher’s LSD test well matched that in SPSS, this might not be enough to prove the figures they got are absolutely being the same! . However, we can calculate the ‘Lower Bound’ and ‘ Upper Bound’ of the 95% confidence interval for counter checking with those on SPSS printout!!

Proving that same figures would be found in using Excel and in using SPSS

Conclusion After activating the ‘Analysis TookPak’ Add-in in Excel, we can have useful statistical tests to use including different Anova tests. We find that, if overall Anova result is significant, we can work further to run Post Hoc Test e.g. Fisher’s LSD Test to find where the mean differences exist, not too difficultly! For One-way Anova, the Excel result has been proved to be consistent with SPSS, even with Unequal Sample Size! Let’s go to Part 2 for Two-way Anova now!!

Part 2 Two-Way Anova

e.g. Fisher’s LSD Test using Excel in aXb factorial Design With Replication - 6 cages each with 4 rats have been used for a Completely Randomized Two-Factors (a x b factorial) With Replication Design Experiment. The 24 rats had been assigned randomly to be subjects for the ‘combinations’ of factor one (Diet A, B, C, D) with factor two (Lighting 1, 2, 3-2 times each). The response is a ‘score’ after the 12 ‘treatments’ e.g. a growing rate in body weight within a certain period of time. Please find any Significant Differences caused by the two factors.

Running the Fisher’s LSD Test in Excel e.g.Two-way Anova aXb Factorial Design With Replication

In Excel with ‘Analysis ToolPak’ Add-In activated, click Data, Data Analysis :-

Choose ‘Two-Factor With Replication’:-

Select Data Area including all Labels :-

A closer look:- Range Including Labels Number of rows of Replication

Output :- Overall Anova Results

Overall Anova Result Lighting ‘MSE’ for Fisher’s LSD Test (Significant) Diet (Significant) Degree of Freedom for MSE Interaction (Not Significant)

For the factor ‘Diet’ the Group Means are:-

Since N is equal = 6 for ALL Groups LSD = t0.05/2,DFW 𝑀𝑆𝑊 1 𝑁1 + 1 𝑁2 = t0.025, 12 10.4167 1 6 + 1 6 = 2.179 X 1.8634 = 4.0603

Any Group Mean Difference > 4.0603 would indicate Significance Group 1 vs Group 2, Group 1 vs Group 4, Group 2 vs Group 3 & Group 3 vs Group 4 are found to be significantly different in their group mean respectively!!

Counter Checking with SPSS Using the same Data Set

The overall results are identical with that in Excel output previously:-

95% confidence interval Lower Bound Upper Bound

Proving that the Excel result are exactly equal to that in SPSS!! Although the Excel result for the Fisher’s LSD test well matched that in SPSS, this might not be enough to prove the figures they got are absolutely being the same! . However, we can calculate the ‘Lower Bound’ and ‘ Upper Bound’ of the 95% confidence interval for counter checking with those on SPSS printout!!

Proving that same figures would be found in using Excel and in using SPSS

Conclusion After activating the ‘Analysis TookPak’ Add-in in Excel, we can have useful statistical tests to use including different Anova tests. We find that, if overall Anova result is significant, we can work further in Excel to run Post Hoc Test e.g. Scheffe’s Test to find where the mean differences exist, not too difficultly! We find that this is not only possible in One-way Anova, but even in Two-way Anova, such as aXb factorial tests!!

Thank You very much!