Statistical Techniques IIEXST7015Treatment Arrangements412321a_Treatments 1Treatment ArrangementsSometimes the treatment simply consists of a list of levels that the investigator is interested in examining. We will term this type of treatment arrangement "a priori" an treatment arrangement. These are often fixed treatment levels that the investigator wants to examine, but they may be random. 21a_Treatments 2Tmt Arrangement (continued)There are several other possibilities. Cross classified (factorial, two-way ANOVA)Like treatments with blocks, two treatments can be cross-classified. Nested treatment arrangement 21a_Treatments 3Tmt Arrangement (continued)A factorial arrangement of treatments occurs when we have two (or more) treatments of interest arranged such that each level of the first occurs with each level of the second. All possible combinations of the two treatments exist. 21a_Treatments 4Tmt Arrangement (continued)Examples - Examine the effect of three dietary supplements (a, b & c) on weight gain for males and females. Each sex gets the same three diets (6 combinations) Examine the effectiveness of three pre-emergence herbicides and four post-emergence herbicides. All of the 12 combinations exist, each treatment may have a null treatment as a control. 21a_Treatments 5Tmt Arrangement (continued)The other type of treatment arrangement is the nested treatment arrangement. Nested treatment arrangements occur when each level of some treatment occurs in combination with some other treatment, but the levels of the second treatment are not the same for each level of the first treatment. 21a_Treatments 6Tmt Arrangement (continued)Examples - Examine the effect of three dietary supplements on weight gain for males and females. Each sex gets three diets, but the diets are different for males (a, b & c) and females (d, e & f). Examine the effectiveness of four post-emergence herbicides on three different crops. The approved post emergence herbicides are not the same for the three crops. 21a_Treatments 7Tmt Arrangement (continued)FactorialA1 A2 A3B1 a1b1 a2b1 a3b1B2 a1b2 a2b2 a3b2B3 a1b3 a2b3 a3b3B4 a1b4 a2b4 a3b421a_Treatments 8Tmt Arrangement (continued)NestedA1A3A2A1B1A1B2A1B3A2B4A2B5A2B6A3B7A3B9A3B821a_Treatments 9Tmt Arrangement (continued)Nested treatment arrangements are not too common. They can occur. For example, if we wanted to test for differences in attendance at State Parks. We choose 4 parks in TX, 5 in LA and 3 in MS. There is no "match" for the parks between states. We could measure attendance on randomly chosen dates and our model would be MODEL Y = STATE PARK(STATE); 21a_Treatments 10Tmt Arrangement (continued)Another example. Suppose we wanted to test for the effectiveness of various commonly used herbicide on major crops in LA by examining dollar value per acre. We choose crops (Cane, Rice, Soy and Corn). We select representative fields at random and treat with an appropriate herbicide. 21a_Treatments 11Tmt Arrangement (continued)Unfortunately, the same herbicides are not used on these crops. Corn and Cane are grasses, and the herbicides target "broadleaf" plants. Soybean is a broadleaf plant, so it requires different herbicides. Rice is grown in water and requires special herbicides. So, each crop has it's own suite of herbicides. MODEL Y = CROP HERB(CROP); 21a_Treatments 12Tmt Arrangement (continued)Factorial designs are VERY common, popular and highly recommended. This treatment arrangement also has some unique properties and interpretations (especially interactions)We will concentrate on this treatment arrangement. 21a_Treatments 13Treatment InteractionsThe one really different thing about treatments is that we are interested in them (as opposed to blocks and nested error terms). We may want to test the individual levels. This will be our major topic following treatment arrangements. We are also likely to be interested in the INTERACTION! 21a_Treatments 14Tmt Interactions (continued)This is new and VERY important. Block & treatment interactions are "error", and not of interest. However, treatment interactions measure how consistent one treatment is across the levels of another. This is interesting and important. It cannot be ignored. 21a_Treatments 15Tmt Interactions (continued)Look at the table below. What value belongs in the missing cell? T2 a T2 b T2 cT1 a 3 5 7T1 b 6 8 10T1 c 2 ???? 6T1 d 5 7 921a_Treatments 16Treatment Interactions (continued)The missing value is 4!!! How did you know? T2 a T2 b T2 c Mean EffectT1 a 3 5 7 5 -1T1 b 6 8 10 8 2T1 c 2 4.00 6 4 -2T1 d 5 7 9 7 1Mean 4 6 8 6Effect -2 0 221a_Treatments 17Tmt Interactions (continued)Could it be 16?T2 a T2 b T2 c Mean EffectT1 a 3 5 7 5 -2T1 b 6 8 10 8 1T1 c 2166 8 1T1 d 5 7 9 7 0Mean 4 9 8 7Effect -3 2 121a_Treatments 18Tmt Interactions (continued)Could it be 1?T2 a T2 b T2 c Mean EffectT1 a 3 5 7 5 -0.75T1 b 6 8 10 8 2.25T1 c 216 3 -2.75T1 d 5 7 9 7 1.25Mean 4 5.25 8 5.75Effect -1.75 -0.5 2.2521a_Treatments 19Tmt Interactions (continued)Of course it can be any value it wants to be. There are no restrictions. However, if it is any value other than 4, then there is an interaction. If we plot the data and there is no interaction, the lines connecting the means should be parallel. 21a_Treatments 20Tmt Interactions (continued)No interaction. 0246810T1a T1b T1c T1dT2 aT2 bT2 c21a_Treatments 21Tmt Interactions (continued)If an interaction is present the lines are not parallel.0246810T1a T1b T1c T1dT2 aT2 bT2 c21a_Treatments 22Tmt Interactions (continued)And may even cross. 0246810T1a T1b T1c T1dT2 aT2 bT2 c21a_Treatments 23Tmt Interactions (continued)So how do we interpret an interaction? If there is no interaction the behavior of the treatments is consistent. The means increase and decrease by the same amount. If there is an interaction, increases and decreased in the means are unpredictable and cannot be foreseen by the main effects. 21a_Treatments 24Tmt Interactions (continued)Of course, in practice no lines are ever EXACTLY parallel. The means never increase and decrease by EXACTLY the same amount. So we need a statistical test to determine if the departure is statistically meaningful; if the interaction is "significant". No problem. We make the interaction a source in our model and test it. 21a_Treatments 25Tmt Interactions (continued)But note one key factor. Blocks had interactions with treatments. We calculated those, and tested