# Section 3: Summary And Software Solution

## Summary

Two-way analysis of variance allows you to examine the effect of two factors simultaneously on the average response. The interaction of these two factors is always the starting point for two-way ANOVA. If the interaction term is significant, then you will ignore the main effects and focus solely on the unique treatments (combinations of the different levels of the two factors). If the interaction term is not significant, then it is appropriate to investigate the presence of the main effect of the response variable separately.

## Software Solutions

### Minitab

General Linear Model: yield vs. fert, irrigation

 Factor Type Levels Values fert fixed 4 100, 150, 200, C irrigation fixed 4 A, B, C, D
 Analysis of Variance for Yield, using Adjusted SS for Tests Source DF Seq SS Adj SS Adj MS F P fert 3 1128272 1128272 376091 12.76 0.000 irrigation 3 161776127 161776127 53925376 1830.16 0.000 fert*irrigation 9 2088667 2088667 232074 7.88 0.000 Error 64 1885746 1885746 29465 Total 79 166878812 S = 171.653 R-Sq = 98.87% R-Sq(adj) = 98.61%
 Unusual Observations for yield Obs yield Fit SE Fit Residual St Resid 4 2390.00 2700.20 76.77 -310.20 -2.02 R 28 2250.00 2646.00 76.77 -396.00 -2.58 R 35 4250.00 3327.60 76.77 922.40 6.01 R R denotes an observation with a large standardized residual. Grouping Information Using Tukey Method and 95.0% Confidence irrigation N Mean Grouping A 20 3120.60 A B 20 3040.05 A C 20 352.85 B D 20 129.55 C Means that do not share a letter are significantly different. Grouping Information Using Tukey Method and 95.0% Confidence fert N Mean Grouping 150 20 1797.90 A 200 20 1749.95 A 100 20 1592.55 B C 20 1502.65 B Means that do not share a letter are significantly different. Grouping Information Using Tukey Method and 95.0% Confidence fert irrigation N Mean Grouping 200 A 5 3381.00 A 150 B 5 3327.60 A 100 A 5 3232.20 A 150 A 5 3169.00 A 200 B 5 3097.00 A C B 5 3089.60 A C A 5 2700.20 B 100 B 5 2646.00 B 150 C 5 623.80 C 100 C 5 340.60 C D 200 C 5 338.00 C D 200 D 5 183.80 D 100 D 5 151.40 D C D 5 111.80 D C C 5 109.00 D 150 D 5 71.20 D Means that do not share a letter are significantly different.

### Excel

 Anova: Two-Factor With Replication SUMMARY Bcontrol B100 B150 B200 Total AA Count 5 5 5 5 20 Sum 13501 16161 15845 16905 62412 Average 2700.2 3232.2 3169 3381 3120.6 Variance 35700.2 4679.2 11167.5 40930 87716.57 AB Count 5 5 5 5 20 Sum 15448 13230 16638 15485 60801 Average 3089.6 2646 3327.6 3097 3040.05 Variance 5839.8 76917.5 269901.3 7432.5 139929.4 AC Count 5 5 5 5 20 Sum 545 1703 3119 1690 7057 Average 109 340.6 623.8 338 352.85 Variance 351.5 2525.8 1079.7 6782.5 37326.03 AD Count 5 5 5 5 20 Sum 559 757 356 919 2591 Average 111.8 151.4 71.2 183.8 129.55 Variance 1485.2 4135.3 997.7 1510.7 3590.366 Total Count 20 20 20 20 Sum 30053 31851 35958 34999 Average 1502.65 1592.55 1797.9 1749.95 Variance 2069464 1977134 2317478 2359637 ANOVA Source of Variation SS df MS F p-value F crit Sample 1.62E+08 3 53925376 1830.164 5.98E-62 2.748191 Columns 1128272 3 376090.7 12.76408 1.23E-06 2.748191 Interaction 2088667 9 232074.2 7.876325 1.02E-07 2.029792 Within 1885746 64 29464.78 Total 1.67E+08 79