Factorial กรณ ม interaction ให ทำ one way

The within-subjects, or repeated measures, design involves administering the various level of independent variables to the same sample of individuals. Because we are able to account for regularities stemming from the participants, we can account for and remove that source of variation from our error term. This reduction in error results in increased statistical power.

Assumptions

We will need to check two assumptions, just as we had for the one-way within-subjects ANOVA.

1. Normality of DV within each combination of levels of IVs
2. Sphericity or equality of variance across difference scores.

A Factorial Within-Subjects Example

Let’s start off with a 3x2 within-subjects design example. By refering to our design as “3x2” indicates that we have two independent variables. The first has 3 levels and the second has 2 levels. The within-subjects name indicates that each participant will receive all levels of all IVs (and have a dependent variable score associated with each administration).

Here’s the set up. A sensation and perception psychologist is investigating the impact of lighting color (natural, green, and red) and intensity (bright and dim) on flavorfulness ratings of vegetables (100 point scale). The researcher invites 5 children to participate in six conditions. The split-plot table is presented in Table 1.

Table 1

Split Plot Design of Example

Color Natural Green Red Intensity Dim P1 = 69 P2 = 69 P3 = 69 P4 = 73 P5 = 74 P1 = 84 P2 = 84 P3 = 84 P4 = 88 P5 = 89 P1 = 31 P2 = 39 P3 = 39 P4 = 35 P5 = 45 Bright P1 = 79 P2 = 79 P3 = 81 P4 = 80 P5 = 84 P1 = 68 P2 = 70 P3 = 71 P4 = 72 P5 = 71 P1 = 52 P2 = 62 P3 = 63 P4 = 56 P5 = 64

Our 3x2 design could be conceptualized as a 3x2x5 design because we are analyzing the variability in flavor scores due to our participants. We will be removing this source of variance rather than examining it for statistical significance. Therefore we do not participant in our description.

Order of Analyses

With our dat set laid out, we can think about the types and number of effects we will want to test in our within-subjects ANOVA. We will want to test for each main effect and for the interaction effect.

Determining Number of Effects

Include all Main Effects.

There will be one main effect for each independent variable. In the Vegetable Flavor example, we have two IV: color and light.

Find Unique Combinations of IV for Interaction Effects.

Eliminate and interaction effects that contain the same factors as another interaciton. Color x Light is the same as Light x Color, so there is only one interaction effect.

The Factorial Within-Subjects ANOVA

When we’re done with checking our assumptions, we run the ANOVA. The ANOVA always precedes the post hoc analyses because it controls for the increased Type I error that accompanies multiple testing.

We’ll run through the SPSS procedure to produce the factorial within-subjects ANOVA using the repeated measures general linear model in the next section, for now we’ll review the interpretation steps. Table 2 is the ANOVA table that results from the repeated measures GLM for our example.

Table 2

Within-Subjects ANOVA Table

Source Sum of Squares df Mean Square F Sig. Color 5368.067 1.06 5044.375 199.556 .000 Error(color) 107.600 4.257 25.278 – – Intensity 213.333 1 213.333 51.200 .002 Error(Intensity) 16.667 4 4.167 – – Color*Intensity 1786.067 1 893.033 1448.162 .000 Error(Color*Intensity) 4.933 4 .617 – –

Note. Degrees of freedom for Color and color*intensity adjusted for sphericity violation using Greenhouse-Geisser correction.

You should notice a few things. First, as indicated by the table note, there was a violation of the assumption of sphericty an that the degrees of freedom were adjusted using the Greenhouse-Geisser correction. Second, there are error terms associated with each effect. That is, we are required to calculated a separate demoninator for each main effect and for the interaction effect.

The need for separate error terms stems from how are grouping our sources of variance. Recall that the error term in our F-statistics represents variability in individual scores about the group means for which we cannot account. In a within-subjects design, we are calculating how the participants’ scores vary from their means and remove that variability from our error term. The reason why this leads to different error terms is because we will have different sets of scores involved in each effect.

Interpreting Effects

Table 1 reveals a significant interaction effect so we will focus our interpreation around that effect. Let’s look to the interaciton plot to guide our interpretation. Figure 1 is a line chart of the effect of color and intensity on reported vegetable flavor.

Figure 1

Interaction Line Graph of Color and Intensity on Vegetable Flavor

Note.Error bars represent 95% CI.

This looks like a busy chart but let’s look for patterns and violations of patterns to guide our interpretation. The first pattern I notice is the nice straight line for the dim lighting conditon. That is, the flavor rating is highest for the green colored light, lowest for the red colored light, and in the middle for natural lighting. I’ll use that straight line as my reference as I examine what is happening in the bright lighting condition.

Recall that a significant interaction tells us that something in the relationship between one IV and the DV changes when we apply the levels of the other IV. As such, we should be looking for where our relationship (i.e., the straight line in the dim condition) changes.

The bright lighting condition does not show the same straight line but rather the highest flavor rating is for natural light rather than green light. Importantly, however, notice that the flavor rating for the green colored light is lower than that of the green colored light in the dim condition just as the red colored light leads to a lower rating than that of the red colored light in the dim condition. Stated another way, we would expect the natural colored light in the bright condition to behave in a similar way to that of the natural colored light in the dim condition (i.e., produce an average flavor rating), but it does not. This deviation from expectation is what is driving our interaction effect and thus we should focus or write-up on that feature.

Reporting Post Hoc Analyses

To corroborate our interpration of the interaction plot, we’ll want some statistics. Just as we had in the one-way within-subjects ANOVA, we will utilize the Bonferroni correction. We do have the option of performing simple effects tests as well. If we were to perform them, I would suggest that we narrow in on the interesting changing happening in the natural light conditions. However, given that we perform the Bonferroni correction for multiple post hoc paired samples t-tests, it makes sense to jump to those corrections now. Table 3 provides the means and confidence intervals for flavor rating at the interaction of the IVs.

Table 3

Means and Bonferroni-adjusted Confidence Intervals

Color Intensity Mean 95% CI LL 95% CI UL Green Bright 70.800 67.708 73.892 Dim 80.600 78.025 83.175 Natural Bright 85.800 82.708 88.892 Dim 70.400 68.517 72.283 Red Bright 37.800 31.324 44.276 Dim 59.400 52.972 65.828

Note.Confidence intervals adjusted using Bonferroni correction.

With these means and confidence intervals, we can provide a succinct description of the interaction.

“The interaction plot in Figure 22 suggests that a trend of flavor ratings decreasing from green light, to natural light, to red light in the dim condition was different for the bright condition. Although the flavor ratings in green and red lights are reliable lower in the bright than dim condition (see Bonferroni-corrected 95% CI in Table 5), the trend reverses for the natural light condition. That is ratings are reliably higher higher in the bright condition than the dim conditon for natural light (see Table 5).”