Need advice applying a logistic regression to model school data

[lcary]

To determine whether schools are increasing the amount of students meeting or
exceeding standards, I obtained a MEAP database reporting the amount of students that
scored in the met or exceed standards range during years 2005-2009, for grades 3-5. I then
graphed the data for using numbers 1 through 5 representing the years since the data
started in the x-axis and and the number of students that met or exceeded standards in the y axis.

I am trying to look at progress over time not just as a linear increase, but as something that might speed up then slow down. In particular, any measurement like % at-or-above-basic-proficiency is limited to between 0% and 100% seems to apply here. A simple model for how it might grow in time would be

f(t) = 1 / (1 + b*e^(-a*t) )

-I tried graphing this for simple values like b=1, a=1 on the range t=-2 to t=+2)
Notice it starts out concave-up on the left, and then turns to be concave-down for larger values of t.

I am trying to ask are schools on the concave-up portion of the curve, or the concave-down portion?

I think I could get a partial answer by including a t^2 term in a regression model, then look at the sign of its coefficient--though there are some dangers, like if the coefficient is positive we might be on the downslope of the parabola, with scores decreasing in time! By the way, I am using t=5 meaning the year 2005.
Overall, I am trying to use logistic regression but I am lost on how to try to apply a non linear program of some sort to find a good model.



any advice?

 

[fresh_42]

If you drop the linearity requirement, then something must replace it. You could e.g. develop a hypothesis, i.e. a curve which you suspect to be a good model, then test it with some standard test, or simply analyse the mean distances of your data. Student's t-distribution might be a good point to start with.