2024-02-05
Which of the following statements is true about the value -0.834 in our regression table?
It is the estimate of the sample intercept
It is the estimate of the population intercept
It is the estimate of the sample slope
Because -0.834 corresponds to the “Age” row of the regression table, this is the slope of our fitted regression line
This means \(\widehat\beta_1 = -0.834\)
\(\widehat\beta_1\) is the estimate of the population slope
\(-0.834\) is just the realized value (the result of fitting the population model)
We read \(\widehat\beta_1 = -0.834\) as: the estimate of the population slope is -0.834
While speaking, I might say “coefficient estimate.” If I am saying estimate, then I mean the population estimate
The following are required parts of the interpretation
Units of Y
Units of X
Mean/average/expected before Y when discussing intercept
Mean/average/expected before difference, increase, or decrease when discussing coefficient for continuous covariate
Confidence interval
Intercept: For someone 0 years old, the average peak exercise heart rate is 214.233 beats per minute (95% CI: 204.918, 223.548).
Slope: For every one year increase in age, the peak exercise heart rate is expected to decrease by 0.834 beats per minute (95% CI: -0.982, -0.685).
Population model: \[ E[Y|X] = \beta_0 + \beta_1X \]
What is \(\beta_1\) mean?
\[ \begin{aligned} E[Y|x_1] & = \beta_0 + \beta_1x_1 \\ E[Y|x_2] & = \beta_0 + \beta_1x_2 \end{aligned}\]
\[ \begin{aligned} E[Y|x_1] - E[Y|x_2] & = (\beta_0 + \beta_1x_1) - (\beta_0 + \beta_1x_2) \\ E[Y|x_1] - E[Y|x_2] & = \beta_0 + \beta_1x_1 - \beta_0 - \beta_1x_2 \\ E[Y|x_1] - E[Y|x_2] & = \beta_1x_1 - \beta_1x_2 \\ E[Y|x_1] - E[Y|x_2] & = \beta_1 (x_1 - x_2) \\ \beta_1 & = \frac{E[Y|x_1] - E[Y|x_2]}{x_1 - x_2} \\ \beta_1 & = \frac{E[Y|x_1] - E[Y|x_2]}{1} \\ \beta_1 & = E[Y|x_1] - E[Y|x_2] \end{aligned}\]
Or: we can look at \(\beta_1\) another way: \[ \begin{aligned} \beta_1 & = E[Y|x_1] - E[Y|x_2] \\ \beta_1 & = E\big[ (Y|x_1) - (Y|x_2) \big] \\ \end{aligned}\]
I really appreciated everyone’s perspective!
I definitely learned a few things while reading
Biggest reason why points were lost: the research question was not focused enough
Taking the test multiple times
A lot of us mentioned learning bias, which can definitely be true
Problems with independence between observations
Generalizability
Does it represent our population? When we just say “population,” is there an unsaid assumption on the population we are referring to?
Can we start to narrow the definition of our population to give context to our sample?
Did not intend for us to get focused on the 3 social theories in the article
Minor writing notes
While folks is a great, inclusive word to describe people, it is a little too informal in reports
Do not use “I” or “think” in report
When we talk about our analysis, avoid how “individuals’” scores relate to their other measures.
Important to note that we are not making conclusions about the individual
We are using individual data to make conclusions about the population!
Make sure you articulate the motivation for your research question
If you are interested in it, then there is likely some research discussing the relationship
Contextualize why this is a research question worth exploring
If you want to review your intro, please come to me!
Good sources for report help
Quiz and Lab 1