Exercise Verify that the term on the left of the equation showing the decomposition of "total" deviation in a two-factor experiment.

Transcription

1 Exercise Verify that the term on the left of the equation showing the decomposition of "total" deviation in a two-factor experiment y ijk y = ( y i y ) + ( y j y ) + [( y ij y ) ( y i y ) ( y j y )] + ( y ijk y ij ) equals the sum of the terms on the right. Exercise Refer to the lecture notes on Two-Factor Experiments. Table 3.4 shows the summary table with row and column means followed by the table of interactions, calculated using the simplified formula Treatment mean Row mean Column mean + Overall mean Table 3.4 Mean iron content for three food types cooked in each of three pot types showing marginal summaries and interaction effects (with one entry missing) Summary data Food Type L M V Pot Type A C I Pot Type Interaction effects Food Type L M V A C I Verify the Legumes/Aluminium interaction effect. Calculate the Vegetable/Iron interaction effect. Verify that the row means and column means of the interaction effects are 0 (within rounding error). Exercise Use the Meat main effect and interaction effects shown in Table 3.4 above to reproduce the original iron content values corresponding to using Meat as shown in the original data. Comment on the difference between Meat effect using iron pots and non iron pots. Exercise Refer to the report on interaction between food types and pot types shown in the lecture notes on Two-Factor Experiments, page 7. Prepare a similar report on interaction patterns in term of the changes in food types for each of the pot types. As an aid to visualising the patterns involved, first produce a food types profile plot analogous to the pot type profiles plot in Figure 3.1, lecture notes on Two-Fator Experiments, page 2.

2 Exercise Refer to the process optimisation experiment discussed in Lecture 2_2.pptx, Slide #47ff. Test the statistical significance of and calculate confidence intervals for the Catalyst effect and the by Catalyst interaction effect. Exercise As part of a project to develop a gas cromatograph (GC) method for analysing trace compounds in wine without the need for prior extraction of the compounds, a synthetic mixture of aroma compounds in ethanol-water was prepared. The result of a GC analysis is displayed in the form of a line plot with spikes or peaks corresponding to the presence of different constituents in the compound. The amount of each constituent is represented numerically by the corresponding peak area. The effects of two factors, Injection volume and Solvent flow rate, on GC measured peak areas given by the mixture were assessed using a 2 2 factorial design with 3 replicate measurements at each design point. The results are shown in the table that follows. Injection volume. L Solvent flow rate, ml/min Display summary results numerically and graphically. What conclusions do these summaries suggest? (EM, Exercise 5.1, pp ) Review Exercise Hydrogels based on polymers are used as carriers of drugs which are released on being dissolved in water. Polymers are linked chains of monomers and there can be varying degrees of crosslinking between the polymer chains in a hydrogel. Light crosslinking results in a hydrogel system that swells in water. The rate of drug release in water depends on the degree of swelling of the hydrogel in the water. Also, certain polymers behave differently below and above a critical temperature, swelling at lower temperatures and shrinking as the temperature is increased above the critical temperature. An experiment was conducted to investigate and quantify the effects of varying degrees of crosslinking and of changing the temperature from below the critical level to above. Four levels of crosslinking were chosen, 1.5%, 3.0%, 7.5% and 15.0%. The temperature was set at either 25 C or 37 C. The level of swelling was measured in terms of the swelling ratio, SR W W W s d, d

3 where W s = weight of swollen hydrogel after immersion in water for a fixed time, W d = weight of dry hydrogel before immersion. A fully randomised factorial design with three replicates at each factor level combination was set up to evaluate the influence of the two factors. The results are given in the following table. Std. Order Run Order % Crosslinked C Swelling Ratio To get an initial overview of the results, the following graphs were produced.

4 Main Effects Plot for Swelling Ratio Data Means % Crosslinked Mean Interaction Plot for Swelling Ratio Data Means 10 8 % Crosslinked Mean (a) Provide interpretations of these plots in terms of the context of the experiment. An analysis of variance was conducted using Minitab, with the following results. Analysis of Variance for Swelling Ratio, using Adjusted SS for Tests Source DF Seq SS Adj SS Adj MS F P % Crosslinked

5 % Crosslinked* Error Total (b) Assuming all necessary assumptions for interpreting this analysis are satisfied, provide an interpretation of these results. A hydrogel that showed the greatest difference in swelling ratio between the two temperatures was required. In pursuit of this requirement, further Minitab output was produced as follows: Least Squares Means for Swelling Ratio % Crosslinked Mean SE Mean n % Crosslinked* (c) (d) Given that a hydrogel that showed the greatest difference in swelling ratio between the two temperatures was required, how would you estimate this difference? Calculate confidence intervals for the differences for the two "best" hydrogels. How would you assess the significance of the difference between two such differences? Calculate a relevant t-statistic for the difference between the two "best" hydrogels and report your conclusion.