Two different approaches for estimating population parameters from a random sample. The need to choose between these two methods primarily occurs in Minitab's reliability commands and Analyze Variability.
Least squares estimates are calculated by fitting a regression line to the points from a data set that has the minimal sum of the deviations squared (least square error). In reliability analysis, this is plotted on a probability plot which can make interpretation easier.
The likelihood function indicates how likely the observed sample is a function of possible parameter values. Therefore, maximizing the likelihood function determines the parameters that are most likely to produce the observed data. From a statistical point of view, MLE is generally recommended for large samples because it is versatile, applicable to most models and different types of data, and produces the most precise estimates.
In many cases, the differences between the LS and MLE results are minor, and the methods can be used interchangeably. You may want to run both methods and see whether the results confirm one another. If the results differ, you may want to determine why. Otherwise, you may want to use the more conservative estimates or consider the advantages of both approaches and make a choice for your problem. There are some areas where one approach has an advantage over the other:
|
|
LSE |
MLE |
|
Biased |
No |
Yes for small samples but decreases as sample size increases |
|
Estimate Variance |
Larger |
Smaller |
|
P-values |
More precise |
Less precise |
|
Coefficients |
Less precise |
More precise |
|
Censored data |
Less reliable and unusable in extreme cases |
More reliable even in extreme cases |
Use LSE when the sample size is small and censoring is not particularly heavy. Otherwise, MLE estimates are generally preferred. Based on their relative strengths, LSE and MLE can be used together for different parts of the analysis. Use LSE's more precise p-values to select the terms to include in the model and use MLE to estimate the final coefficients.
For some commands, Minitab calculates the scale parameter using an adjusted ML estimate, which is simply the sample standard deviation (for the normal distribution) or the sample standard deviation of the transformed data (for the Box-Cox and Johnson transformations and the lognormal distribution).