Estudio 1. Abuse of Technology in Adolescence and its Relation to Social and Emotional

Two-way ANOVA is similar to one-way ANOVA except that two factors (or two independent variables) are analyzed. For example,

in the study described in Presenting Problem 2, Gonzalo and colleagues (1996) wanted to know if a difference existed in insulin sensitivity depending on thyroid level or body mass index (BMI). They defined hyperthyroidism as an increase in serum thyroxine, free thyroxine index, and suppressed serum thyroid-stimulating hormone levels; overweight was defined as BMI > 25 (body mass index is calculated as weight in kilograms divided by height in meters squared, kg/m²). Raw data are given in Table 7-8. In this example, both factors are measured at two levels on all subjects and are said to be crossed.

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Table 7-7. Multiple comparisons for free T₄ (pmol/L) using NCSS.

Bonferroni (All -Pairwise) Multiple Comparison Test

Response: FT₄ Term A: Group

Alpha=0.050 Error Term=S DF=70 MSE=3.521327 Critical Value=2.4529

Group Count Mean Different from Groups

1 20 13.55 3

2 18 14 3

3 35 15.88571 1, 2

Newman–Keuls Multiple-Comparison Test

Response: FT₄ Term A: Group

Alpha=0.050 Error Term=S DF=70 MSE=3.521327

Group Count Mean Different from Groups

1 20 13.55 3

2 18 14 3

3 35 15.88571 1, 2

Scheffe's Multiple-Comparison Test

Response: FT₄ Term A: Group

Alpha=0.050 Error Term=S DF=70 MSE=3.521327 Critical Value=2.5011

Group Count Mean Different from Groups

Because two factors are analyzed in this study (thyroid level and BMI), each measured at two levels (hyperthyroid vs controls and overweight vs normal weight), 2 × 2 = 4 treatment combinations are possible: overweight hyperthyroid subjects, overweight controls, normal weight hyperthyroid subjects, and normal

weight controls. Three questions may be asked in this two-way ANOVA:

1. Do differences exist between hyperthyroid subjects and controls? If so, the means for each treatment combination might resemble the hypothetical values given in Table 7-9A, and we say a difference exists in the main effect for thyroid status. The null hypothesis for this question is that insulin sensitivity is the same in hyperthyroid subjects and in controls (µ_H = µ_C).

2. Do differences exist between overweight and normal weight subjects? If so, the means for each treatment combination might be similar to those in Table 7-9B, and we say a difference occurs in the main effect for weight. The null hypothesis for this question is that insulin sensitivity is the same in overweight subjects and in normal weight subjects ( µ_O = µ_N).

3. Do differences exist owing to neither thyroid status nor weight alone but to the combination of factors? If so, the means for each treatment combination might resemble those in Table 7-9C , and we say an interaction effect occurs between the two factors. The null hypothesis for this question is that any difference in insulin sensitivity between overweight hyperthyroid subjects and overweight controls is the same as the difference between normal weight hyperthyroid subjects and normal weight controls (µ_OH – µ_OC = µ_{N H} – µ_{N C})

1 20 13.55 3

2 18 14 3

3 35 15.88571 1, 2

Tukey–Kramer Multiple-Comparison Test

Response: FT₄ Term A: Group

Alpha=0.050 Error Term=S DF=70 MSE=3.521327 Critical Value=3.3864

Group Count Mean Different from Groups

1 20 13.55 3

2 18 14 3

3 35 15.88571 1, 2

Source: Observations, used with permission, from Woeber KA: Levothyroxine therapy and serum free thyroxine and free triiodothyronine concentrations. J Endocrinol Invest 2002; 25: 106–109. Table produced using NCSS 2001, a registered trademark of the Number Cruncher Statistical System; used with permission.

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Table 7-8. Insulin sensitivity for women in different groups.

Normal Thyroid Hyperthyroid

Normal Weight Overweight Normal Weight Overweight

0.97 0.76 0.56 0.19

0.88 0.44 0.89 0.11

0.66 0.48 0.55 0.13

0.52 0.39 0.66 0.21

This study can be viewed as two separate experiments on the same set of subjects for each of the first two questions. The third question can be answered, however, only in a single experiment in which both factors are measured and more than one observation is made at each treatment combination of the factors (ie, in each cell).

The topic of interactions is important and worth pursuing a bit further. Figure 7-4A is a graph of hypothetical mean insulin

sensitivity levels from Table 7-9A for hyperthyroid and control, overweight and normal weight women. When lines connecting means are parallel, no interaction exists between the factors of thyroid

status and weight status, and the effects are said to be additive. If the interaction is significant, however, as in Table 7-9C, the lines intersect and the effects are called multiplicative. Figure 7-4B illustrates this situation and shows that main effects, such as thyroid status and overweight status, are difficult to interpret when significant interactions occur. For example, if the interaction is significant, any conclusions regarding increased insulin sensitivity depend on both thyroid status and weight; any comparison between women with hyperthyroidism and controls depends on the weight of the subject. Although this example illustrates an extreme interaction, many statisticians recommend that the interaction be tested first and, if it is significant, main effects not be tested.

0.38 1.10 0.11 0.32

0.71 0.19 0.27 0.01

0.46 0.19 0.56

0.29 0.19 0.80

0.68

0.96

0.97

N 11.00 8 8 6

Mean 0.68 0.47 0.55 0.16

SD 0.25 0.32 0.26 0.10

All Control Women All Hyperthyroid Women

N 19 14

Mean 0.59 0.38

SD 0.29 0.28

All Overweight Women All Normal Weight Women

N 14 19

Mean 0.34 0.63

SD 0.29 0.25

Source: Data, used with permission, from Gonzalo MA, Grant C, Moreno I, Garcia FJ, Suarez AI, Herrera-Pombo JL, et al: Glucose tolerance, insulin secretion, insulin sensitivity and glucose effectiveness in normal and overweight hyperthyroid women. Clin Endocrinol 1996; 45: 689–697.

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The calculations in two-way ANOVA are tedious and will not be illustrated in this book. They are conceptually similar to the calculations in the simpler one-way situation, however: The total variation in observations is divided, and sums of squares are determined for the first factor, the second factor, the interaction of the factors, and the error (residual), which is analogous to the within-group sums of squares in one-way ANOVA.