Robert Martin turned 55 in 1991. Earlier in that same year, the Westvaco Corporation, which makes paper products, decided to downsize. They ended up laying off roughly half of the 50 employees in the engineering department where Martin worked, including Martin. Later that year, Martin went to court, claiming that he had been fired because of his age. A major piece of evidence in Martin’s case was based on a statistical analysis of the relationship between the ages of the workers and whether they lost their jobs.
Part of the data analysis presented at his trial concerned the ten hourly workers who were at risk of layoff in the second of five rounds of reductions. At the beginning of Round 2 of the layoffs, there were ten employees in this group. Their ages were 25, 33, 35, 38, 48, 53, 55, 55, 56, 64. Three were chosen for layoff: the two 55-year-olds (including Martin) and the 64-year old. (See Martin v. Westvaco case details.)
Create a side-by-side stemplot of the ages of the employees who were retained and laid off. Use the table below where the middle column shows the stem (tens digit) and you fill in the leaves (ones digits) on either side.
Based on your stemplot, comment on whether these data provide preliminary evidence that the layoff decisions were not made at random. What aspects of the age distributions support or contradict this claim?
The data provide preliminary evidence that the layoffs may not have been random. If the three employees were chosen completely at random from the ten, we would expect a mix of ages. However, all three laid-off employees (55, 55, 64) were among the four oldest workers, with an average age of about 58 years compared to about 41 years for the retained employees. This clustering of older ages in the laid-off group seems unlikely to occur by chance alone, suggesting possible age discrimination. However, we need to conduct a formal statistical analysis to determine whether this pattern is statistically significant or could reasonably occur by random chance.
If the layoff decisions were made completely at random, we would expect the ages in the laid-off group and the retained group to look similar - they should overlap considerably and have similar centers and spreads. There shouldn’t be a systematic difference where one group is noticeably older or younger than the other.
One way would be to write the 10 ages on slips of paper, mix them thoroughly, and randomly draw 3 slips to represent the laid-off employees. The remaining 7 would be the retained employees. Alternatively, you could number the employees 1-10 and use a random number generator to select 3 numbers. Each random selection would give you one "could have been" outcome showing what the layoffs might have looked like if they were truly random.
The ages of the laid-off individuals were 55₁, 55₂, and 64. Which other sets of 3 ages from the 10 would you consider as or more extreme? Briefly justify your choices.
The observed outcome had the three oldest workers laid off (two 55-year-olds and the 64-year-old). More extreme outcomes would be: (55₁, 56, 64), (55₂, 56, 64), and (55₁, 55₂, 56). These three combinations all involve laying off 3 of the 4 oldest workers.
There are 4 outcomes as or more extreme (including the observed): the observed outcome (55₁, 55₂, 64) plus the three from the previous question. So the p-value = 4/120 = 0.0333 or about 3.3%. This means if layoffs were random, there would be only a 3.3% chance of seeing a combination as extreme as what was observed. This provides strong evidence that the layoff decisions were not made at random and may have discriminated based on age.
Approach 2: Approach 1 focuses on who was laid off and who was not, but does not fully consider how old they were. Rather than ignoring this information, we could instead focus on the difference in the ages assigned to the two groups.
Calculate the mean age for the laid-off group (55, 55, 64) and the mean age for the retained group (25, 33, 35, 38, 48, 53, 56), then find the difference.
Identify any outcomes that have a larger difference in the average age between those laid off/not. Are the same outcomes still the most extreme? Does the p-value change?
Only the combinations (55₁, 56, 64) and (55₂, 56, 64) have a larger difference in means (17.33 years each). The combination (55₁, 55₂, 56) has a smaller difference (13.04 years). So when using difference in means as our statistic, we have 3 outcomes as or more extreme (the observed plus the two with difference 17.33), giving p-value = 3/120 = 0.025 or 2.5%, which is slightly stronger evidence than the 3.3% from Approach 1.
This is an observational study, not a randomized experiment, so we cannot establish a cause-and-effect relationship. While the data provide evidence that the layoffs were not random, we cannot definitively conclude that age was the cause of the layoff decisions - there could be other confounding variables. Additionally, this analysis only applies to these 10 specific workers in Round 2; we cannot necessarily generalize to all employees or all rounds of layoffs without additional data.
The distribution is roughly centered around 0 (since under random assignment, we’d expect no systematic difference between the groups). The distribution is not symmetric because the group sizes are unequal (7 vs. 3) and because the dataset is small, creating granularity in the possible values for the differences.
Although we might suspect the distribution to be centered at zero, the graph is not expected to be symmetric because the group sizes are unequal and we subtracted laid-off – retained. Also, the data set is small so there will be “granularity” between the possible values for the difference. In fact, other possible choices of statistics (e.g., difference in medians) would look even “stranger.” But those details are less important, we can still use these values to determine a p-value.
In this analysis, you assumed a null hypothesis – the lay-offs are made completely at random – and you considered all 120 possible assignments of 7 people to be retained and 3 people to be laid off. (You could have flipped a coin for each employee, but that would lay off 50% in the long-run rather than fixing 3 to be laid off.) Of these 120 possible assignments, we want to know how many are “at least as extreme” as the observed result. If we focus on the outcomes where 3 of the 4 oldest people end up in the laid-off group, only the 55₁, 56, 64 and 55₂, 56, 64 combinations have a difference in means (17.33 years, which can happen two ways) larger than what we observed (16.86). The 55₁, 55₂, 56 combination and the 53, 56, 64 combination don’t have quite as large a difference in means (13.04 and 16.38 respectively). So this finds an exact p-value = 3/120 = 0.025, which tells us that if these decisions had been made completely at random, there is a 2.5% chance we would see a combination of ages at least as extreme as the actual outcome as measured by the difference in group means. This is moderate evidence that the decision-making process was not at random. If you do find this evidence compelling, you must keep in mind the observational nature of the data; we cannot draw any cause-and-effect conclusions about the reasons for the non-random lay-off decisions.
In the rest of this chapter, you will focus on comparing groups on a quantitative response variable. The reasoning process of statistical significance remains the same – how often would random chance give me a result at least as extreme? Your analysis will still focus on two key ideas: How were the groups formed (e.g., random assignment vs. random sampling), or are we modelling hypothetical randomness? And choice of statistic for comparing the two groups. With large sample sizes we won’t be able to list out all the outcomes so we will focus more on simulation and mathematical models. The latter will also again allow us to use a convenient confidence interval formula. Once you have a p-value or confidence interval, you interpret them in the same way as in previous chapters, while also considering the limitations of the study design.
Suppose the following individuals were laid off: 25, 35, 38. What would be the exact p-value if the alternative hypothesis was that there was a bias against the younger employees? Explain.
