ISCAM: Investigating Statistical Concepts, Applications, and Methods

To summarize the distribution of a categorical variable, we can simply count how many are in each category and make a bar graph to display the results, one bar for each outcome, with heights representing the number of observations in each category, separating the bars to indicate distinct categories.

InfantData = read.table(url("https://www.rossmanchance.com/iscam3/data/InfantData.txt"), header=TRUE)

InfantData = read.table(file.choose(), header=TRUE)

 choice
1    Helper
2    Helper
3    Helper
4    Helper
5    Helper
6  Hinderer

choice
  Helper Hinderer 
  14        2

Clearly a majority/more than half of the infants chose the helper toy in this sample of 16 infants. But does that convince us that infants in general are more likely to pick the helper toy in the long run? In other words, what is the probability that an infant will choose the helper toy?

Model assumption: Note we are assuming each infant has the same probability of picking the helper toy, we just don’t know the value of that probability.

So that leaves us with two explanations for the majority we observed:

Our analysis approach is going to be to assume the second explanation is true (similar to how in a legal trial we assume a defendant is innocent), and then see whether our data are consistent or inconsistent with that assumption. To do this, we need to investigate the values we expect to see for the number choosing the helper toy when 16 infants are equally choosing between the two toys. As you saw with the Random Babies (Investigation B), we can simulate the outcomes of a random process to help us determine which outcomes are more or less likely to occur.

For a 50-50 simulation model, we can flip a fair coin. We can arbitrarily define “heads” to be choosing the helper toy and “tails” to be choosing the hinderer toy. We will repeat the random process 16 times to represent the 16 infants, and we will count how many times we flip heads, representing an infant choosing the helper toy. The chart below shows this mapping of the real world, which we saw one instance of, and the simulation model, which we can easily repeat many times. Keep in mind that in the simulation model, we know the probability of heads is 0.50.

We really need to simulate this hypothetical random selection process hundreds, preferably thousands of times. This would be very tedious and time-consuming with coins, so let’s turn to technology.

Returning to our legal trial analogy, if you decide that the observed “data” is unlikely to occur by chance alone, you are going to “reject” the assumption of “innocence” (and say we have evidence the defendant is guilty). If you decide the data/evidence is not unusual by chance alone, then you “fail to reject” that assumption (and we say we don’t have evidence the defendant is guilty — we aren’t proving the defendant innocent, just that the evidence is not inconsistent with that assumption, “not guilty”).

So based on these simulation results, we would say the data (14 helper choices out of 16 trials) is unusual under the assumption that infants genuinely have no preference and are choosing blindly when presenting the toys. This evidence convinces us that “There is something to the theory that infants are genuinely more likely to pick the helper toy (for some reason)” is the more believable explanation for why so many of the infants in this study picked the helper toy over the hinderer toy. We haven’t proven this is true, but based on the strong majority these researchers saw, even for this small sample size of 16, we would consider the evidence convincing (“beyond a reasonable doubt”) that, in the long run, the probability of choosing the helper toy in this random process is greater than 0.5. Because the researchers controlled for other possible explanations for the observed preference results like color and handedness, we will conclude that there is convincing evidence that infants really do have a genuine preference for the helper toy over the hindering toy.