Question

QUESTION 42 Make up an example of a study that uses a 2 x 2 factorial design, and fill in a table of cell means that would show no main effects and no interaction effect. (Do not use an example from your textbook class lectures, or your classmates.) Explain the pattern of the cell means you created within the context of your example. T T T Arial Path: p Click Save and Submit to save and submit. Click Save All Annvers ro save all arinwers

Answer 1

Two by two factorial design contains 2 factors each at 2 levels.

SAS output of the data.

EXAMPLE (A 2 x 2 balanced design): A virologist is interested in studying the effects of a 2 different culture media (M) and b-2 different times (T) on the growth of a particular virus. She performs a balanced design with n-6 replicates for each of the 4 M*T treatment combinations. The N - 24 measurements were taken in a completely randomized order. The results: THE DATA TOTALS Medium1 Medium 2 T=2 12 21 23 20 25 24 29 T hours 22 28 2626 25 27 18 37 38 3531 29 30 hours39 38 36 34 33 35 T= 12 | y11-= 140 | y12.-156 | yi..-296 у.1.-363 y.2.-348 711 у MEANS i = Level of T kObservation number ikkth observation from the ith J = Level of M M =2 22. = 亚-2-29.00 level of T and jth level of M 亚-1. = 30.25 亚--.-29.625 . The effect of changing T from 12 to 18 hours on the response depends on the level of M -For medium 1, the T effect 37.lễ-23.3 - For medium 2, the T effect32 - 26 - . The effect on the response of changing M from medium 1 to 2 depends on the level of T - For T 12 hours, the M effect26 23.3- - For T 18 hours, the M effect32-37.16

If either of these pairs of estimated effects are significantly different then we say there exists a significant interaction between factors M and T. For the 2 x 2 design example: -If 13.นี้ is significantly different than 6 for the M effects, then we have a significant M * T Or. -If 2.6 is significantly different than-5.1ถึ for the T effects, then we have a significant M * T . There are two ways of defining an interacion between two factors A and B: - If the average change in response between the levels o factor A is not the same at all levels of factor B, then an interaction exists between factors A and B - The lack of additivity of factors A and B, or the nonparallelism of the mean profiles of A and B, is called the interaction of A and B When we assume there is no interaction between A and B, we say the effects are additive. An interaction plot or treatment means plot is a graphical tool for checking for potential interactions between two factors. To make an interaction plot, 1. Calculate the cell means for all α-b combinations of the levels of A and B 2. Plot the cell means against the levels of factor A. 3. Connect and label means the same levels of factor B . The roles of A and B can be reversed to make a second interaction plot. Interpretation of the interaction plot: - Parallel lines usually indicate no significant interaction. - Severe lack of parallelism usually indicates a significant interaction. - Moderate lack of parallelism suggests a possible significant interaction may exist Statistical significance of an interaction effect depends on the magnitude of the MSE For smal values of the MSE, even small interaction effects (less nonparallelism) may be significant When an As B interaction is large, the corresponding main effects A and B may have little practical meaning. Knowledge of the A * B interaction is often more useful than knowledge of the main effect. We usually say that a significant interaction can mask the interpretation of significant main effects That is, the experimenter must examine the levels of one factor, say A, at fixed levels ol the other factor to draw conclusions about the main effect of A It is possible to have a significant interaction between two factors, while the main effects for both factors are not significant. This would happen when the interaction plot shows interactions in different directions that balance out over one or both factors (such as an X pattern). This type of interaction, however, is uncommon

ANOVA and Estimation of Effects for a 2x2 Design The GLM Procedure Dependent Variable: growth Sum of Source Model Error Corrected Total 23 793.6250000 DF Squares Mean Square F Value Pr > F 31691.4583333230.496111 45.12 .0001 20 102.1666667 5.1083333 R-Square Coeff Var Root MSE growth Mean 0.871267.6292402.260162 29.6250 Source time mediunm DF Type III SS Mean Square F Value Pr> F 90.0416667 5900416667115.51 2000 1.84 0.1906 18.02 0,0004 19.3750000 9.3750000 time medium 1 92.0416667 92.0416667 Parameter Intercept time 12 time 18 medium 1 medium 2 time medium 121-7.83333333 B 1.84541474 Error t Value Pr> 2.00000000 B 0.9220073734.68.0001 6.00000000 B 1,30490528 0,00000000 B 5.16666667 B 1.30490528 0.00000000 B 4.60 00002 3.96 00008 14200004 1220,00000000 B time medium 181 0,00000000 B 182 0,00000000 B

was Note: The X'X matrix has been found to be singular, and a generalized inverse was used to solve the normal equations. Terms whose estimates are followed by the letter 'B' are not uniquely estimable. B are not uniquely Standard Error t Value Pr>It 29.6250000 0.46135368 6421 |く0001 1<0001 10.75<0001 1.35 0.1906 1.35 0.1906 time-12 medium-1-1.9583333 0.461353684.24 0.0004 4.24 0.0004 4.24 0.0004 time-18 medium-2-1.9583333 0.461353684.24 0.0004 time-12 4.9583333 0.4613536810.75 4.9583333 0.46135368 10.75 0.6250000 0.4613536 0.6250000 0.46135368 time-1 time-12 medium-21.9583333 0.46135368 time-18 medium-1 1.9583333 0.46135368

Fit Diagnostics for growth 25 30 35 25 35 0.20 0.25 0.30 Predicted Value Predicted Value Leverage 35 0.15 0.10 0.05 0.00 30 F 0 25 20 2 1 02 Quantile 20 25 30 35 40 0 5 10 15 20 25 Predicted Value

Level of timeN Level of medium N Mean Std Dev 24.6666667 2.77434131 12 34.5833333 3.28794861 MeanStd Dev 12 30.2500000 7.58137670 12 29.0000000 3.71728151 12 12 24666667 277434151 2

Interaction Plot for growth Distribution of growth Tests for Normality Test p Value 05737 0.1500 Cramer-von Mises W.sq 0.049547 Pr> W.Sq >0.2500 A-Sq 0303234 Pr>A-Sq >0.2500 Shapiro-Wilk 0.140051 Pr> D ime 121 122 182 growth Level of Level of time medium N Mean Std Dev 12.0000000 2.366431

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