I just added an example of simple model construction to my textbook, Statistical Inference for Everyone. It's a process I don't think I've ever seen in an intro stats book, but is common in scientific work. The idea is that you start off with a simple model, collect data, then notice where your simple model breaks, propose a new more complex model, and do the analysis again.

The entire data set I use is here, where I have the mass of US Pennies for several years:

Year

Mass

1960

3.133

1961

3.083

1962

3.175

1963

3.120

1964

3.100

1965

3.060

1966

3.100

1967

3.100

1968

3.073

1969

3.076

1970

3.100

1971

3.110

1972

3.080

1973

3.100

1974

3.093

1989

2.516

1990

2.500

1991

2.500

1992

2.500

1993

2.503

1994

2.500

1995

2.497

1996

2.500

1997

2.494

1998

2.512

1999

2.521

2000

2.499

2001

2.523

2002

2.518

2003

2.520

Single "True" Value Model

One starts this analysis loading the first part (earlier than 1975), and applying a model which states that there is a single "true" value. The best estimate of this value is the sample mean, and the posterior distribution is normal. A plot of this looks like

If you apply it to all the data, you get something that clearly looks ridiculous:

It is then that it makes sense to change the model to a two "true" values model.

Double "True" Value Model

With this model, we have separate means for the pre- and post-1975 data, and can look at the overlap of the credible intervals, or the posterior distribution of the difference, both of which clearly show a statistically significant difference.

Advantages

This approach has several advantages over the typically methods used to teach this topic:

it progresses systematically from simple to complex

it shows the benefits and limitations of the simple models

it connects the procedures of the complex models to the earlier ones, so they don't seem like disjoint unrelated topics.

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