Data Set/Problem Description

Statistics 5372: Experimental Statistics

Homework Due March 1 -- Key

Data Set/Problem Description

A 2-factor fixed effects analysis of variance was run to examine the stability of a drug product stored at 4 storage times (1,3,6, and 9 months) and in 2 labs. The vials were analyzed for mg/mL of the active ingredient and pH at the end of the specified storage times in order to assess stability. A 2-factor ANOVA was run for each variable. There were 3 replicates for each laboratory/storage time combination. The data are given on page 935 of the text.

Variable 1: Active Ingredient (in mg/mL) at End of Storage Period

Key Results of the Analysis

The analysis of variance table (Table 1) for the analysis of mg/mL indicates that there is a significant interaction between lab and storage time (p = .0003). Thus, we do not test for main effects, but instead we compare the cell means. The cell means for both variables are given in Table 2. LSD for comparing cell means at the  level of significance is

The cell means are given in Table 2 and are plotted via interaction plots in Figure 1. In Table 3 we provide the LSD multiple comparison calculations and provide a display of the resulting significant differences. From the interaction plots we get the impression that the difference in lab storage facility makes more difference for 3 and 6 month storage than it does for 1 or 9 month storage. From the multiple comparisons, it can be seen that lab 1 reading at 3 months is significantly higher than all other readings except lab 1 at 1 month. The lowest 3 readings were from both labs at 9 months and lab 2 at 6 months. The lab 2 reading at 3 months was significantly different from all other readings. The 1 month readings from the two labs were not significantly different from each other or from the 1 month reading from lab 1.

In Figure 2 we present side-by-side boxplots of the data as well as a probability plot and histogram of the residuals. The box plots show a fairly consistent variation and suggest that equality of variances is a reasonable assumption. The probability plot and especially the histogram suggest that the normality assumption may be questionable.

Conclusionsin the Language of the Problem

There is very little difference between the two labs after 1 and 9 months. However, at 3 and 6 months, lab 2 storage resulted in significantly less active ingredient suggesting that lab 2 is less desirable. Also apparent is the general decline in active ingredient present as storage time increases as would be expected. It is interesting to note that the curious increase in active ingredient from 1 month to 3 months in lab 1 is not significant.

Variable 1: pH

Key Results of the Analysis

The analysis of variance table (Table 4) for the analysis of pH shows that there is not a significant interaction between lab and storage time (p = .4038) while there is a significant difference between labs (p < .0001) and among times (p = .0001). The interaction plot is shown in Figure 3 which agrees with the lack of significant interaction in that no distinct patterns of interaction appear. Since there are only two labs, there is no need to perform multiple comparisons on labs. However, these are shown in Table 4 where it can be seen that lab 2 storage results in significantly higher pH readings. The comparisons of times given in Table 4 shows that the pH at 6 months is significantly higher than the other 3 time periods and that the 1 month reading is significantly higher than the 3 month reading.

In Figure 4 we present side-by-side boxplots of the data as well as a probability plot and histogram of the residuals. The box plots show some dramatic differences among variances. However, a definitive assessment of the equality of variances is not available because of the small number of replicates (i.e. n = 3). The histogram of the residuals looks fairly bell-shaped but the probability plot does suggest possible non-normality.

Conclusions in the Language of the Problem

In general, storage in lab 2 seems to consistently result in higher pH readings than for lab 1. The storage time comparisons of pH readings give confusing results in that the 6-month reading is significantly higher than all others. There was no consistent pattern of decline or increase in pH in time as might have been expected. If lower pH is desirable (?) then these results along with those for the mg/mL of active ingredient suggest that lab 1 is the preferred storage facility.

Appendices:

A. Tables and Figures Cited in the Report

Table 1. 2-Factor ANOVA - Ex 15.41, page 935 -- mg/mL Data

The GLM Procedure

Dependent Variable: mgml

Sum of

Source DF Squares Mean Square F Value Pr > F

Model 7 0.46740000 0.06677143 27.30 <.0001

Error 16 0.03913333 0.00244583

Corrected Total 23 0.50653333

R-Square Coeff Var Root MSE mgml Mean

0.922743 0.165090 0.049455 29.95667

Source DF Type III SS Mean Square F Value Pr > F

time 3 0.29376667 0.09792222 40.04 <.0001

lab 1 0.09126667 0.09126667 37.32 <.0001

time*lab 3 0.08236667 0.02745556 11.23 0.0003

Table 2. Cell Means for Ex.6, page 550

------time=1 lab=1 ------

The MEANS Procedure

Variable Mean Std Dev