Testing the effect of two factors at different levels (two treatments). Examples: Yield of a crop varying with fertilizer type and seedling used.

Transcription

1 ANOVA TWO-WAY

2 Testing the effect of two factors at different levels (two treatments). Examples: Yield of a crop varying with fertilizer type and seedling used. Gas mileage depends on gas additive and tires used Impact of a drug on hemoglobin level depending on the dosage and the gender of patient. Color density depends on light level (lux) and paper Sales varies with advertisement budget and market size

3 Note that in the first three examples, the factors are qualitative, in the fourth one factor (lux) is quantitative and the other qualitative, and in the last example, both factors are quantitative. Also, we observe that the two factors may have different levels.

4 Analyzing the effect of oxygen concentration and sugar on fermentation ( as measured by the amount of ethanol. Four possible oxygen levels and two possible sugar levels are considered. Each combination of oxygen-sugar levels is a treatment. So, there are 8 treatments in this experiment. Each treatment consists of two observations. For the sake of simplicity, let us call the factors of Oxygen as Factor A and factors of Sugar as Factor B. Data: Ethanol concentration (in millimoles per mg.) for each treatment are given below: There are k levels of factor A, l levels of factor B, m observations for each treatment, so that there are klm observations in our sample.

5 Oxygen Sugar Galactose Glucose , , , , , , , , 0.01

6 Questions: Is there any significant interaction between the Oxygen and the type of Sugar to determine ethanol concentrations? Testing for interaction effect. If there is no interaction,

7 If there is no interaction.. (a) do the oxygen levels by themselves affect the ethanol concentrations significantly? That is, are the mean ethanol concentrations at the four oxygen levels significantly different, over both levels of sugar? -- Testing for the main effect of oxygen; Main effect A

8 If there is no interaction.. contd. (b) do the sugar levels by themselves affect the ethanol concentrations significantly? That is, are the mean ethanol concentrations at the two sugar levels significantly different, over all levels of Oxygen? -- Testing for the main effect of Sugar; Main effect B

9 ethanol(mmpmg) Oxygen Sugar In our example, is there any interaction effect? Look at the graphs of summary measures. Note that as the level of Oxygen changes from 1 to 4, the change in the average response is approximately the same whether the sugar is at level 1 or 2. The two lines joining the average responses at these two levels are approximately parallel. So, we do not expect much interaction effect Sugar Oxygen 1 Galactose Glucose , , , , , , , , 0.01 Individual Value Plot of ethanol(mmpmg) vs Oxygen, Sugar

10 Another way

11 Another way

12 SCORE Consider the accompanying graph of mean responses for two factors each at two levels. Here, as Factor A changes from level L to S, the mean response changes in different ways depending on whether Factor B is at level L or S. So, clearly there is interaction present and in this case, it would not be useful to test for main effects of A and B Lecture EXAM Individual Value Plot of SCORE vs EXAM, Lecture L S L S L S

13 Since there is no significant interaction effect in our example, we could now look for the presence of main effects. That is we could ask if the mean responses are significantly different among the levels of Factor A and among the levels of Factor B.

14 In terms of sums of squares, this translates to SST = SSA + SSB+SSAB + SSE with degrees of freedom for SST = klm -1; degrees of freedom for SSA = k-1; degrees of freedom for SSB = l-1; degrees of freedom for interaction SSAB = (k-1)(l-1); and degrees of freedom for error = kl(m-1). The data is entered in three columns one for the levels of Factor A, one for the levels of Factor B, and the third with the response values. The sample Minitab worksheet is as follows: Oxygen Sugar ethanol(mmpmg) Choose STAT ANOVA TWO WAY Enter the columns for response, and factors; Check the boxes for display means Click the Graphs button; Check Individual Value Plot and if need be Box plot. Click OK twice.

15 Two-way ANOVA: ethanol(mmpmg) versus Oxygen, Sugar Source DF SS MS F P Oxygen Sugar Interaction Error Total S = R-Sq = 74.12% R-Sq(adj) = 51.47%