Transformations of Data Example 1 illustrated the use of two-w...

Question

Transformations of Data Example 1 illustrated the use of two-way ANOVA to analyze the sample data in Table 12-3. How are the results affected in each of the following cases?

b. Each sample value is multiplied by the same nonzero constant.

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So, first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Suppose you conduct a two-way ANOVA on a data set and then multiply every value in the data set by a constant factor of 2. What effect does this transformation have on the F statistics and P values in the ANOVA results? Awesome. So it appears for this particular problem, we're ultimately trying to determine what effect will this, rather these conditions set to us by the prom itself, what would the conditions of this prom. Effect wise, what effect will this transformation have on the F statistics and P values in this specific ANOVA results, which, as we should recall, ANOVA is to denote analysis of variation for variants or variation. So analysis of variance. That's what ANOVA stands for. Awesome. So with that in mind, now that we know what we're ultimately trying to solve for, let us recall and note that when we multiply every value, which in this case we're multiplying every value by a constant factor or two. So when we multiply every value in the data set by a constant factor, which in this case, 2, It will have a specific effect, or rather specific effects. On the F statistics and P values in this two way ANOVA, which once again ANOVA is analysis of variance. So now, with that in mind, our first step to solve this particular problem is we need to focus our attention on what effect it has on the F statistics. So as you should recall and note, the F statistic. In ANOVA is a ratio of the variance between groups to the variance within groups, and it's computed as, as we should recall, and use the following equation. So F equals the variance between groups, which I'll call VBG for short, so F is equal to the variance between groups divided by the variance within groups. Which I'll call VWG. So once again, F is equal to the variance between groups divided by the variance within groups. And as we should recall and note, if we multiply. Every value in the data set by a constant factor such as 2 in this case, both the numerator, the variance between groups, and the denominator, the variance within groups will be affected by the square of that constant. So since variance is based on the square of the data values, multiplying all values by 2 will scale both the between group and within group variances by a factor of 2 to 2, which is equal to 4. So that would mean that the new F. Statistic, which I'll call NFS for short. So the new Fstatistic will be equal to 4 multiplied by the variance between groups, divided by 4 multiplied by the variance within groups. Fantastic. And this will be all equal to the original Fstatistic value, which I'll call OFS for short. So to quickly recap, so the new Fstatistic will be equal to 4 times the variance between groups, divided by 4 multiplied by the variance within groups. Which is going to be equal to ultimately the original Fstatistic value. So, in conclusion, the F statistic will remain unchanged by multiplying all values by a constant value. So that will mean, ultimately, the ratio of variances will stay the same. So now, our second step is we need to determine what the effect will be on P values. So as we should recall a note, the P value in ANOVA is determined by comparing the calculated F statistic to a reference distribution, which is based on the degrees of freedom and the assumptions of the test. And since the F statistic does not change when you multiply all the data points by a constant value, that will mean, as a result, the P value will also remain unchanged. So therefore, in conclusion, the P value is not affected by multiplying all the values by a constant factor. Therefore, without a doubt, it will remain the same. So that will mean our final answer for this particular problem, without a doubt, will have to be. That the F statistic and the P values both remain unchanged, and that's it. We've officially solved for this problem. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Transformations of Data Example 1 illustrated the use of two-way ANOVA to analyze the sample data in Table 12-3. How are the results affected in each of the following cases?

b. Each sample value is multiplied by the same nonzero constant.

Key Concepts

Two-Way ANOVA

Data Transformation

Effect of Constant Multiplication

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