Explaining Maximum Likelihood Estimation

Here are two techniques that I've found particularly useful in teaching about maximum likelihood estimation.

(1) What is the difference between "probability theory" and "statistical inference"?

Most students have been exposed to the binomial distribution, and those who have not quickly get the general idea when asked to consider flipping a possibly biased coin some fixed number of times, say 5. I display a table such as that below, in which the binomial distribution is used to calculate the probability of zero through five heads in 5 tosses of a coin for 5 possible values of a bias parameter:

Bias of coin towards Heads
Heads .1 .3 .5 .7 .9
0
1
2
3
4
5
.59
.33
.07
.01
.00
.00
.17
.36
.31
.13
.03
.00
.03
.16
.31
.31
.16
.03
.00
.03
.13
.31
.36
.17
.00
.00
.01
.07
.33
.59

I tell the class that if we are dealing with probability theory, then we look at columns of the table. For example, if we have a fair coin, we look at the column labelled ".5". But as statisticians, we deal with situations in which we have already performed an experiment, and want to make an inference about what state of nature probably produced the outcome. Therefore, statisticians look at the rows of the table.

We identify the row corresponding to the outcome that was observed. If we observe four heads, then the numbers in that row of the table show the likelihood function, calculated for these five possible values of the true underlying proportion. Since any value of the proportion between zero and one is possible, we must imagine a similar table with more and more values; or in the limit, a graph showing the shape of the function.

Since the numbers in a row do not add up to one (or, considering the true case of infinite values, do not integrate to one), these are not probabilities; that is why the different name "likelihood" was used by Ronald Fisher to describe these numbers. We can see that higher values of the likelihood correspond to underlying proportions that are more likely to produce the observed outcome. We naturally define the maximum likelihood estimate to be the number for which this likelihood is the largest. Since the biggest likelihood with 4 heads in this table is associated a bias of .7, then the maximum likelihood estimate is that the coin bias is .7.

(2) Maximizing a function is like climbing a hill with a bucket on your head.

There are frequently many parameters to be estimated at once in a general function. We must discover the estimates for all the parameters that simultaneously maximize the likelihood. A frequently-used analogy for finding maxima of general functions is climbing a hill.

Students who are told this don't see what is so difficult about this; they think that all you have to do is look at the hill, and see where the highest point is. What they don't understand is that in maximizing a general function, you don't usually get to see the whole hill at once, but only a small part of it. So I describe the process as one of being blindfolded and led to some point, and then having the blindfold removed while a bucket is placed over your head. The bucket allows you to see a small area around you, but you can't see the whole landscape. Based on what you see "nearby" by looking down, you have to decide what direction to move in, and how far. You are then transported to that spot (blindfolded again), and are then allowed to repeat the process. You must decide when you have reached the top of the hill.

If students can visualize the process this way, they can see more of the complexities involved, including possible multiple hill tops, some of which are higher than others.

David Rindskopf
Educational Psychology
City University of New York Graduate School

Explaining maximum likelihood estimation.Rindskopf D. … Rasch Measurement Transactions, 1998, 12:3 p. 645.


Please help with Standard Dataset 4: Andrich Rating Scale Model



Rasch Publications
Rasch Measurement Transactions (free, online) Rasch Measurement research papers (free, online) Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Applying the Rasch Model 3rd. Ed., Bond & Fox Best Test Design, Wright & Stone
Rating Scale Analysis, Wright & Masters Introduction to Rasch Measurement, E. Smith & R. Smith Introduction to Many-Facet Rasch Measurement, Thomas Eckes Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr. Statistical Analyses for Language Testers, Rita Green
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Journal of Applied Measurement Rasch models for measurement, David Andrich Constructing Measures, Mark Wilson Rasch Analysis in the Human Sciences, Boone, Stave, Yale
in Spanish: Análisis de Rasch para todos, Agustín Tristán Mediciones, Posicionamientos y Diagnósticos Competitivos, Juan Ramón Oreja Rodríguez

To be emailed about new material on www.rasch.org
please enter your email address here:

I want to Subscribe: & click below
I want to Unsubscribe: & click below

Please set your SPAM filter to accept emails from Rasch.org

www.rasch.org welcomes your comments:

Your email address (if you want us to reply):

 

ForumRasch Measurement Forum to discuss any Rasch-related topic

Go to Top of Page
Go to index of all Rasch Measurement Transactions
AERA members: Join the Rasch Measurement SIG and receive the printed version of RMT
Some back issues of RMT are available as bound volumes
Subscribe to Journal of Applied Measurement

Go to Institute for Objective Measurement Home Page. The Rasch Measurement SIG (AERA) thanks the Institute for Objective Measurement for inviting the publication of Rasch Measurement Transactions on the Institute's website, www.rasch.org.

Coming Rasch-related Events
May 26 - June 23, 2017, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
June 30 - July 29, 2017, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com
July 31 - Aug. 3, 2017, Mon.-Thurs. Joint IMEKO TC1-TC7-TC13 Symposium 2017: Measurement Science challenges in Natural and Social Sciences, Rio de Janeiro, Brazil, imeko-tc7-rio.org.br
Aug. 7-9, 2017, Mon-Wed. In-person workshop and research coloquium: Effect size of family and school indexes in writing competence using TERCE data (C. Pardo, A. Atorressi, Winsteps), Bariloche Argentina. Carlos Pardo, Universidad Catòlica de Colombia
Aug. 7-9, 2017, Mon-Wed. PROMS 2017: Pacific Rim Objective Measurement Symposium, Sabah, Borneo, Malaysia, proms.promsociety.org/2017/
Aug. 10, 2017, Thurs. In-person Winsteps Training Workshop (M. Linacre, Winsteps), Sydney, Australia. www.winsteps.com/sydneyws.htm
Aug. 11 - Sept. 8, 2017, Fri.-Fri. On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com
Aug. 18-21, 2017, Fri.-Mon. IACAT 2017: International Association for Computerized Adaptive Testing, Niigata, Japan, iacat.org
Sept. 15-16, 2017, Fri.-Sat. IOMC 2017: International Outcome Measurement Conference, Chicago, jampress.org/iomc2017.htm
Oct. 13 - Nov. 10, 2017, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
Jan. 5 - Feb. 2, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
Jan. 10-16, 2018, Wed.-Tues. In-person workshop: Advanced Course in Rasch Measurement Theory and the application of RUMM2030, Perth, Australia (D. Andrich), Announcement
Jan. 17-19, 2018, Wed.-Fri. Rasch Conference: Seventh International Conference on Probabilistic Models for Measurement, Matilda Bay Club, Perth, Australia, Website
April 13-17, 2018, Fri.-Tues. AERA, New York, NY, www.aera.net
May 25 - June 22, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
June 29 - July 27, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com
Aug. 10 - Sept. 7, 2018, Fri.-Fri. On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com
Oct. 12 - Nov. 9, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
The HTML to add "Coming Rasch-related Events" to your webpage is:
<script type="text/javascript" src="http://www.rasch.org/events.txt"></script>

 

The URL of this page is www.rasch.org/rmt/rmt1237.htm

Website: www.rasch.org/rmt/contents.htm