Chapter 16.
Two-Way Analysis of Variance for Independent Samples
Part 3

• Example 2.

Our second example uses the same research setting as in Example 1 and, as you will see in a moment, almost the same data. Now that we have laid the groundwork for the two-way independent-samples ANOVA, the computational side of this example will be quick and easy. The tricky part will the interpretation of the results, once we have them.

First a reminder of the setting: To test the separate and mutual effects of two drugs, A and B, on physiological arousal, researchers randomly and independently sorted 40 laboratory rats into four groups of 10 subjects each. Each group received a certain dosage of drug A (zero units or 1 unit) and a certain dosage of drug B (zero units or 1 unit). The dependent variable was a standard measure of physiological arousal. One of the groups served as a control, receiving only an inert placebo containing zero units of A and zero units of B.

The following table shows the consequent measures of physiological arousal for each subject in each of the four groups. If you have a sharp eye and a keen memory, you will see that all of the values listed for groups 1, 2, and 3 are the same as in Example 1. The only difference is in the listing for group 4, whose members receive 1 unit of A and 1 unit of B.

 rawdata B 0 units 1 unit A 0units 20.4 17.4   20.0 18.4   24.5 21.0   19.7 22.3   17.3 23.3 20.5 26.3   26.6 19.8   25.4 28.2   22.6 23.7   22.5 22.6 1 unit 22.4 19.1   22.4 25.4   26.2 25.1   28.8 21.8   26.3 25.2 17.5 13.6   16.9 12.4   16.4 18.3   13.6 19.1   16.1 20.5

As the procedure follows the same format we worked through in Example 1, we will zip through it with only a minimum of commentary. (Click here if you would like a printable summary of the raw data and summary values for this example.)

Summary Values from Preliminary Number-Crunching
 summarydata B 0 units 1 unit rows A 0units Ng1=10 ∑Xg1=204.3 ∑X2g1=4226.3 Ng2=10 ∑Xg2=238.2 ∑X2g2=5741.4 Nr1=20 ∑Xr1=442.5 1 unit Ng3=10 ∑Xg3=242.7 ∑X2g3=5961.35 Ng4=10 ∑Xg4=164.4 ∑X2g4=2763.66 Nr2=20 ∑Xr2=407.1 columns Nc1=20 ∑Xc1=447.0 Nc2=20 ∑Xc2=402.6 NT=40 ∑XT=849.6 ∑X2T=18692.7

Means and Graph of Group Means
 B 0 units 1 unit rows A 0units Mg1=20.43 Mg2=23.82 Mr1=22.13 1 unit Mg3=24.27 Mg4=16.44 Mr2=20.36 columns Mc1=22.35 Mc2=20.13 MT=21.24

Preliminary SS Values
 B 0 units 1 unit A 0units SSg1=52.45 SSg2=67.48 1 unit SSg3=71.02 SSg4=60.92 SST=647.2

 SSwg = SSg1 + SSg2 + SSg3 + SSg4 = 52.45 + 67.48 + 71.02 + 60.92 = 251.87

 SSbg = SST — SSwg = 647.2 — 251.87 = 395.33

 SSrows = (∑Xr1)2Nr1 + (∑Xr2)2Nr2 — (∑XT)2NT = (442.5)220 + (407.1)220 — (849.6)240 = 31.33

 SScols = (∑Xc1)2Nc1 + (∑Xc2)2Nc2 — (∑XT)2NT = (447.0)220 + (402.6)220 — (849.6)240 = 49.28

 SSrxc = SSbg — SSrows — SScols = 395.33 — 31.33 — 49.28 = 314.72

The following table shows (in red) the values of [null]Mg* for each of the four groups, as calculated by the method described in connection with Example 1. As before, the observed means of the groups (20.43, 23.82, etc.) appear in black. The graphs below the table show the observed group means in comparison with the pattern that would be expected if there were zero interaction between the row and column variables. As you can see from both the table and the graphs, there is a substantial difference between the observed and the expected.

 means B 0 units 1 unit rows A 0units 20.4323.24 23.8221.02 Mr1=22.13 1 unit 24.2721.47 16.4419.25 Mr2=20.36 columns Mc1=22.35 Mc2=20.13 MT=21.24
 observed expected

Degrees of Freedom
 degrees offreedom in general for thepresentexample Total dfT = NT—1 40—1=39 within- groups (error) dfwg = NT—rc 40—(2)(2)=36 between- groups dfbg = rc—1 (2)(2)—1=3 rows dfrows = r—1 2—1=1 columns dfcols = c—1 2—1=1 interaction dfrxc = (r—1)(c—1) (2—1)(2—1)=1

MS Values
 MSrows = SSrowsdfrows MScols = SScolsdfcols MSrxc = SSrxcdfrxc = 31.331 = 49.281 = 314.721 = 31.33 = 49.28 = 314.72

 MSerror = SSwgdfwg = 251.8736 = 7.0

F-ratios
 Frows = MSrowsMSerror Fcols = MScolsMSerror Frxc = MSrxcMSerror = 31.337.0 = 49.287.0 = 314.727.0 = 4.48 = 7.04 = 44.94 with df=1,36 with df=1,36 with df=1,36

Here again is the sampling distribution of F for df=1,36, along with the corresponding tabular portion of Appendix D. The critical values for the .05 and .01 levels of significance are F=4.11 and F=7.40, respectively.

Figure 16.2. Sampling Distribution of F for df=1,36

 dfdenomi-nator df numerator 1 2 3 36 4.117.40 3.265.25 2.874.38

In brief, all three of the effects are significant: the main effects for rows and columns (Frows=4.47, Fcols=7.03) both beyond the .05 level, and the large interaction effect (Frxc=44.88) far beyond the .01 level. This is the easy part of the analysis. Now for the trickier part, which is to figure out what it means.

Once again, the fundamental meaning of the significant row and column effects is that the difference between the two row means (22.13 vs 20.36) and the difference between the two column means (22.35 vs 20.13) each reflect something more than mere random variability. But note what happens when you simply take these two differences at face value, without reference to the very substantial interaction effect. As indicated by the following plots of the row and column means, the "obvious" inference is that drugs A and B both decrease arousal.

Examine the data in closer detail, however, and you will see that this conclusion is quite the opposite of what is really happening. For group 1, which receives only the inert placebo, the mean level of arousal is 20.43. For group 2, which receives 1 unit of B and none of A, it is higher: 23.82. For group 3, which receives 1 unit of A and none of B, it is also higher: 24.27. The complication comes with group 4, which receives 1 unit each of A and B; for here the effects of the two drugs in combination are not merely additive.

 B 0 units 1 unit A 0units Mg1=20.43 Mg2=23.82 1 unit Mg3=24.27 Mg4=16.44
Indeed, they are the very opposite of additive: A and B in combination produce a mean arousal level of 16.44, substantially lower than the 20.43 observed with the inert placebo group. Presented separately, each of the two drugs increases arousal, while in combination they interact to produce a marked decrease in arousal.

 Although the above graph captures the texture of the interaction effect in this particular example, the more generic format is the one shown earlier (now adjacent), in which the individual group means are plotted across their respective rows. The fundamental generic meaning of a significant interaction effect is that the difference between Interaction Effect (i) the observed pattern of group means and (ii) the pattern that would be expected if the combined effects of the row and column variables were merely additive reflects something more than mere random variability.

ANOVA Summary Table
 Source SS df MS F P between groups 395.19 3 rows 31.33 1 31.33 4.48 <.05 columns 49.28 1 49.28 7.04 <.05 interaction 314.72 1 314.72 44.96 <.01 within groups(error) 251.87 36 7.0 TOTAL 647.2 39

End of Chapter 16, Part 3.
Return to Top of Chapter 16, Part 3
Go to Chapter 16, Part 4

 Home Click this link only if the present page does not appear in a frameset headed by the logo Concepts and Applications of Inferential Statistics