Howdy! I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help. Today we're going to learn how to recognize the effect of data transformations on two-way ANOVA results. Here's our problem statement: The accompanying data shows a sample of pulse rates in beats per minute that were categorized with two factors: age bracket in years and gender. The data were used to illustrate the way to use a two-way ANOVA. How are the results affected in each of the following cases? A) the same constant is added to each sample value; B) each sample value was multiplied by the same non-zero constant; C) the format of the table is transposed so that the row and column factors are interchanged; D) the first sample value in the first cell is changed so that it becomes an outlier.
OK, so the first part of this problem, Part A, wants us to consider if we transform the data by adding the same constant to each data value, how does that affect the ANOVA results? Will the test statistic change? Will the P-value change? Well, if you consider the way that the test statistic is actually calculated, you come to understand that you know it's based on differences between individual data values and the mean value and then you square those differences. So you know adding the same constant to each sample value is going to shift everything on your number line. But the differences between the mean value and each individual value is still going to be the same. So you know when you square those differences you're going to get the same numbers coming out. So that's not going to change your results at all. Nice work!
Now Part B wants us to consider a data transformation in which we multiply each sample value by the same non-zero constant. Well, this again is going to not have an effect, because when you multiply each data value with the same non-zero constant, in essence what you're doing is you're keeping the proportions of the squares of the differences the same. So again, that's not going to change anything. Nice work!
Next, Part C wants us to consider if we transpose the row and column factor so that their data are interchanged. So the row data becomes the column data, the column data becomes the row data. Well again, you know the calculation for your values there in your ANOVA table. And the division that you make to get this, the test statistic, it's kind of similar to what you see with the linear correlation coefficient where it doesn't matter if you swap the X and Ys; you're still going to get the same value out for R. It's the same sort of thing here. You're still going to get the same values that you use to divide to get your test statistic out. So the test statistic is going to stay the same. And of course, if the test statistic stays the same, so is the P value; that's not going to change either. So yeah, you can swap data around all you want. That's not going to change anything. Nice work!
And now the last part, Part D says, "Choose the correct answer if we just change the first data value in the first cell so that it becomes an outlier." Well, OK, now we're going to see some changes here because, see, before in each of these three instances we were changing all of the data the same way. Now we're going to change just one data value. We'll leave the rest of it alone. Now that's good. Now we're going to see a difference with that. And just as we saw with wide swings in our linear correlation coefficient with the introduction of an outlier in our data set, we're going to see the same thing here. If you put an outlier in your data set, that's going to radically shift the value of the test statistic. And hence also the P-value is going to change too.
So let's see what we got here. Yeah, this one looks good. "Both the test statistic and P-value will most likely change because outliers can dramatically affect and change the results of an ANOVA." Yeah, that's going to be the one they want, but let's check the other options just to make sure. "The P-value and mean will only change by a very small amount because ANOVA is robust against outliers." That's definitely not true. "The P-value will be approximately 1 minus the previous P-value." Where in the world did that come from? I don't know. OK, so I'm pretty sure with the answer we got up here. Nice work!
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