In sum,
given the
differences in composition between the two exams along with possible changes in student effort
over the
time period, it is extremely difficult to determine what one should expect to see under the best of circumstances.
Alpha: The alpha of a mutual fund describes the
difference between a fund's actual return
over a
period of
time and its expected return,
given the fund's level of risk.
Then, instead of throwing out the data as hopelessly compromised and starting the experiment
over with these factors corrected, you (a) do a study estimating how miscalibrated, how defective and how improperly located your instruments were and apply adjustments to all past data to «correct» the improper reading, (b) you do a study to estimate the effect of the external factors at the
time you discover the problem and apply adjustments to all past data to «correct» the effects of the external factors even though you have no idea what the effect of the external factor actually was for a
given instrument at the
time the data was recorded, because you only measured the effect years later and then at only some locations, (c) you «fill in» any missing data using data from other instruments and / or from other measurements by the same instrument, (d) you do another study to determine how best to deal with measurements from different instruments
over different
time periods and at different locations and apply adjustments to all past data to «correct» for
differences between readings from different instruments
over different
time periods at different locations.
Linear regression determines the underlying trend in a dataset
over a
given period as the slope of the unique straight line through the data that minimizes the sum of the squares of the absolute
differences or «residuals» between the data - points corresponding to each
time interval in the data and on the trend - line.