7 Lightweight aggregate asphalt mix has been found to have lower thermal conductiv-

Example 11.7 Lightweight aggregate asphalt mix has been found to have lower thermal conductiv-

  ity than a conventional mix, which is desirable. The article “Influence of Selected Mix Design Factors on the Thermal Behavior of Lightweight Aggregate Asphalt Mixes” (J. of Testing and Eval., 2008: 1–8) reported on an experiment in which var- ious thermal properties of mixes were determined. Three different binder grades were used in combination with three different coarse aggregate contents (), with two observations made for each such combination, resulting in the conductivity data (Wm °K) that appears in Table 11.6.

  Table 11.6 Conductivity Data for Example 11.7 Coarse Aggregate Content ()

  Asphalt Binder Grade

  x j .8400

  Here I5J53 and K52 for a total of IJK 5 18 observations. The results of the analysis are summarized in the ANOVA table which appears as Table 11.7 (a table with additional information appeared in the cited paper).

  CHAPTER 11 Multifactor Analysis of Variance

  Table 11.7 ANOVA Table for Example 11.7 Source

  The P-value for testing for the presence of interaction effects is .414, which is clearly

  larger than any reasonable significance level. Alternatively, f AB 5 1.10 , F .10,4,9 5

  2.69 , so the interaction null hypothesis cannot be rejected even at the largest signifi- cance level that would be used in practice. Thus it appears that there is no interaction between the two factors. However, both main effects are significant at the 5 signif- icance level ( .002 .05 and .000 .05 ; alternatively both corresponding F ratios

  greatly exceed F .05,2,9 5 4.26 ). So it appears that true average conductivity depends

  on which grade is used and also on the level of coarse-aggregate content.

  Figure 11.5(a) shows an interaction plot for the conductivity data. Notice the nearly parallel sets of line segments for the three different asphalt grades, in agreement with the F test that shows no significant interaction effects. True average conductivity appears to decrease as aggregate content decreases. Figure 11.5(b) shows an interaction plot for the response variable thermal diffusivity, values of which appear in the cited arti- cle. The bottom two sets of line segments are close to parallel, but differ markedly from those for PG64; in fact, the F ratio for interaction effects is highly significant here.

  0.86 Asph Gr

  Asph Gr

  Mean 0.81 Mean 2.3

  Agg Cont

  Agg Cont

  (a)

  (b)

  Figure 11.5 Interaction Plots for the Asphalt Data of Example 11.7. (a) Response variable is conductivity. (b) Response variable is diffusivity

  Plausibility of the normality and constant variance assumptions can be assessed by constructing plots similar to those of Section 11.1. Define the predicted (i.e., fitted)

  values to be the cell means: xˆ ijk 5x ij . For example, the predicted value for grade

  PG58 and aggregate content 38 is xˆ 11k 5 (.835 1 .845)2 5 .840 for k 5 1, 2 . The residuals are the differences between the observations and corresponding predicted

  values: x ijk 2x ij . A normal probability plot of the residuals is shown in Figure 11.6(a).

  11.2 Two-Factor ANOVA with K ij .1

  The pattern is sufficiently linear that there should be no concern about lack of normality. The plot of residuals against predicted values in Figure 11.6(b) shows a bit less spread on the right than on the left, but not enough of a differential to be worri- some; constant variance seems to be a reasonable assumption.

  –0.015 –0.010 –0.005 0.000 0.005 0.010 0.015

  Residual

  Fitted Value

  (a)

  (b)

  Figure 11.6 Plots for Checking Normality and Constant Variance Assumptions in Example 11.7

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