Suppose I have the following dataset that contains measurements on different patients (each unique ID is a unique patient) - "x" variables represent covariates and "y" variables represent outcomes:
id x_1 x_2 x_3 x_4 x_5 y_1 y_2 y_3 y_4 y_5 y_6 y_7 y_8 y_9 y_10
1 1 137.71008 83.22509 96.77519 139.199703 -31.05143 20.78144 201.37399 251.993953 343.69970 231.62894 278.26058 63.90400 179.571790 -7.370581 135.52294
2 2 240.57995 76.04897 -21.00893 3.407949 161.11235 115.08062 101.07260 200.581716 112.08013 177.05726 215.56689 145.16217 109.613811 112.834290 37.73935
3 3 66.92601 179.04881 -34.18888 3.106041 225.28584 56.88339 220.26244 3.416566 172.49610 54.01245 11.05028 53.05884 100.335158 77.712706 296.31327
4 4 120.87087 202.61092 176.98454 140.890102 142.51842 -64.06773 -23.43913 169.147210 129.27064 -70.55632 22.39471 30.87697 147.580290 43.288333 111.33670
5 5 56.29673 48.33402 119.25437 45.504666 89.87237 198.60063 181.48716 24.956806 55.75533 101.08906 18.19344 163.20589 5.623714 85.519361 148.23663
6 6 60.13320 156.73718 -10.77539 -79.319398 -47.54688 127.28152 49.17664 126.881145 37.45836 142.68339 103.83295 231.97907 186.168837 235.994598 230.42569
I am interested in fitting the following regression models - I believe this is called "Multiple Regression":
model_1 <- lm(y_1 ~ x_1 + x_2 + x_3 + x_4 + x_5 , data = my_data)
model_2 <- lm(y_2 ~ x_1 + x_2 + x_3 + x_4 + x_5 , data = my_data)
model_3 <- lm(y_3 ~ x_1 + x_2 + x_3 + x_4 + x_5 , data = my_data)
model_4 <- lm(y_4 ~ x_1 + x_2 + x_3 + x_4 + x_5 , data = my_data)
model_5 <- lm(y_5 ~ x_1 + x_2 + x_3 + x_4 + x_5 , data = my_data)
model_6 <- lm(y_6 ~ x_1 + x_2 + x_3 + x_4 + x_5 , data = my_data)
model_7 <- lm(y_7 ~ x_1 + x_2 + x_3 + x_4 + x_5 , data = my_data)
model_8 <- lm(y_8 ~ x_1 + x_2 + x_3 + x_4 + x_5 , data = my_data)
model_9 <- lm(y_9 ~ x_1 + x_2 + x_3 + x_4 + x_5 , data = my_data)
model_10 <- lm(y_10 ~ x_1 + x_2 + x_3 + x_4 + x_5 , data = my_data)
Based on the results of these regression models, suppose I want to see that if each "beta" coefficient associated with "x_1" for each model can be considered statistically significant or not.
I have been told that for such problems, a "correction" such as the Bonferroni Correction is often required to choose a more "conservative" value of "alpha" that will be used in these hypothesis tests. But I struggle to understand why this necessary. For example, if I were to inspect the results of these models:
> model_1
Call:
lm(formula = y_1 ~ x_1 + x_2 + x_3 + x_4 + x_5, data = my_data)
Coefficients:
(Intercept) x_1 x_2 x_3 x_4 x_5
131.15352 -0.09218 -0.03841 0.02782 -0.07882 -0.05099
> model_7
Call:
lm(formula = y_7 ~ x_1 + x_2 + x_3 + x_4 + x_5, data = my_data)
Coefficients:
(Intercept) x_1 x_2 x_3 x_4 x_5
108.07151 -0.06348 0.01031 -0.05734 -0.06381 -0.12770
I can't seem to understand why determining if "-0.09218" is statistically significant - how this somehow is related to whether "-0.06348" is statistically significant. The calculations required to calculate the beta coefficients for "model 1" do not seem to directly affect the calculations required to calculate the beta coefficients for "model 7" - even though both of these models are using the same dataset. This being said, why should testing the statistical significance of coefficients from "model 1" require somehow "correcting" for "model 2" and vice versa?
Can someone please help me understand as to why in this problem that I have outlined, a "correction factor" is required for individual comparisons and why all individual hypothesis tests for these models need to be compared at a lower value of "alpha" - even though all these individual comparisons seem independent?
I can understand if someone was interested in determining if all "x_1" coefficients for all models were JOINTLY statistically significant why a correction factor might be needed - but I am having trouble understand why testing whether " -0.09218" is statistically significant by itself still required a correction factor. Where exactly is the "Family" in this case?
Thanks!