High School and Beyond Data
. describe
Contains data from hsb.dta
obs: 7,185
vars: 7 27 Oct 2020 21:35
────────────────────────────────────────────────────────────────────────────────────
storage display value
variable name type format label variable label
────────────────────────────────────────────────────────────────────────────────────
female byte %8.0g female
ses float %9.0g socioeconomic status
mathach float %9.0g math achievement
size int %8.0g school size
sector byte %8.0g Catholic vs. Public
schoolid float %9.0g School ID
mathgroup float %9.0g math group (Hi / Lo)
────────────────────────────────────────────────────────────────────────────────────
Sorted by:
. summarize
Variable │ Obs Mean Std. Dev. Min Max
─────────────┼─────────────────────────────────────────────────────────
female │ 7,185 .5281837 .4992398 0 1
ses │ 7,185 .0001434 .7793552 -3.758 2.692
mathach │ 7,185 12.74785 6.878246 -2.832 24.993
size │ 7,185 1056.862 604.1725 100 2713
sector │ 7,185 .4931106 .4999873 0 1
─────────────┼─────────────────────────────────────────────────────────
schoolid │ 7,185 5277.898 2499.578 1224 9586
mathgroup │ 7,185 .5000696 .5000348 0 1
Histogram of SES
. histogram ses, scheme(michigan)
(bin=38, start=-3.7579999, width=.16973684)
. graph export myhistogram.png, width(1000) replace
(file /Users/agrogan/Desktop/newstuff/categorical/multilevel/myhistogram.png written
> in PNG format)
Generate Mean SES Per School
. bysort schoolid: egen meanses = mean(ses) // generate mean ses per school
. summarize ses meanses
Variable │ Obs Mean Std. Dev. Min Max
─────────────┼─────────────────────────────────────────────────────────
ses │ 7,185 .0001434 .7793552 -3.758 2.692
meanses │ 7,185 .0001434 .4135432 -1.193946 .8249825
. twoway scatter ses meanses, msize(vsmall) ///
> title("SES and Mean SES Are Correlated") sub("But Not Equivalent") ///
> scheme(michigan)
. graph export myscatter.png, width(1000) replace
(file /Users/agrogan/Desktop/newstuff/categorical/multilevel/myscatter.png written i
> n PNG format)
Visualizing The Idea Of A Random Intercept
. twoway (function y = logistic(x), range(-5 5)) /// first school; random intercept
> 0
> (function y = logistic(x + 1), range(-5 5)) /// second school; random intercept 1
> (function y = logistic(x - 1), range(-5 5)), /// third school; random intercept -1
> title("Three Hypothetical Schools") ///
> sub("With Different Random Intercepts") ///
> legend(order(1 "random intercept 0" 2 "random intercept +1" 3 "random intercept -1
> ")) ///
> scheme(michigan)
. graph export myMLM.png, width(1000) replace
(file /Users/agrogan/Desktop/newstuff/categorical/multilevel/myMLM.png written in PN
> G format)
Developing Some Notation
Our notation for logistic regression model is:
\[ \ln \Big( \frac{p(\text{outcome})}{1 - p(\text{outcome})} \Big) = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + ... \]
which after exponentiating both sides, and some rearrangement, can be written:
\[ p(\text{outcome}) = \frac{e^{\beta_0 + \beta_1 x_1 + \beta_2 x_2 + ...}}{1 +e^{\beta_0 + \beta_1 x_1 + \beta_2 x_2 + ...}} = \]
\[ F(\beta_0 + \beta_1 x_1 + \beta_2 x_2 + ...) \]
where \(F(z) = \frac{e^z}{1 + e^z}\), which is the logistic distribution.
So in adapting this notation for the multilevel context, we are ultimately going to write the notation for the multilevel logistic regression model as:
\(p(\text{outcome} | \text{unique intercept for each unit}) = F(\beta_0 + \beta_1 x_1 + \beta_2 x_2 + ... + u_{0j})\)
