Probabilistic models for measurement Copenhagen 2010
The Rasch model as a statistical model:
Erling B. Andersens contributions to the theory of Rasch
models
Karl Bang Christensen
Svend Kreiner
Univ. of Copenhagen
presentation availabe from http://staff.pubhealth.ku.dk/~kach/.
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Probabilistic models for measurement Copenhagen 2010
Rasch ModelOrdinal items X1, . . . , XI measuring latent variable θ
Pr(Xi = 1|θ) =exp(θ − βi)
1 + exp(θ − βi)=
ξδi
1 + ξδi(1)
data X = (Xvi), values θ1, . . . , θN , scores Xv. =∑
i Xvi.
MATHEMATICS
RM
STATISTICS PHILOSOPHY
Expression of a comparison (θ − βi). Developed 1955-1960.
Rasch. Probabilistic Models for some Intelligence and Attainment Tests. Copenhagen:Danish National Institute for Educational Research 1960
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Probabilistic models for measurement Copenhagen 2010
Erling B. Andersen
studied with Rasch 1961-1963.
first Danish graduate with formal degree in mathematical statistics 1963
gold medal from Univ. of Copenhagen for work on Rasch model 1965
first elected chairman of the Danish Soc. for Theor. Statistics 1971 (?)
defended his doctoral thesis on conditional inference 1973.
took over Rasch’s chair at Dep. of Statistics, Univ. of Copenhagen 1974.
editor of Scand. Journal of Statistics 1974-1986.
remembered as an excellent lecturer
his 1980 book is a classic still well worth reading today.
(?): Reasons for forming society at that time: SJS + Rasch’s 70th birthday in sept. 1971.
Andersen. Discrete stat. models with social science appl. Amsterdam: North-Holland, 1980.
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Probabilistic models for measurement Copenhagen 2010
Consistency - better estimates with more observations
(Xj)j=1,...,n normal variables, E(Xj) = µ and V (Xj) = σ2
MLE µ̂ = 1n
∑j xj consistent: µ̂→ µ as n→∞
MLE fails under random sampling from non-identical distr.
(Xij)i=1,...,I,j=1,...,n, with E(Xij) = µi and V (Xij) = σ2
µ̂i consistent, but σ̂ is inconsistent: σ̂ → k−1k σ as n→∞.
General formulation by Neyman & Scott
Neyman & Scott. Consistent estimates based on partially consistent observations.Econometrika 1948, 16:1-32.
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Probabilistic models for measurement Copenhagen 2010
Conditional inference
Conditioning yields a sequence of independent random variables.
X1|T1 , X2|T2 , . . .
Using the conditional distribution given the sum scores yields
CML. The estimates are consistent and asymptotically normally
distributed.
Exact test known since 1925. Results on UMPU tests 1937.
Bartlett. Properties of suff. and stat. tests. Phil. Trans. Royal Soc. A 1937, 160:268-82.
Fisher. Statistical Methods for Research Workers. Edinburgh: Oliver & Boyd, 1925
Neyman. Outline of a Theory of Statistical Estimation Based on the Classical Theory ofProbability. Phil. Trans. Royal Soc. A 1937, 236:333-80.
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Probabilistic models for measurement Copenhagen 2010
Writing δ = (δ1, . . . , δI), X = (X1, . . . , XI), x = (x1, . . . , xI) it
turns out that we can do estimation using
LC(δ) =n∏
v=1
LC(δ;xv) (2)
where
LC(δ;x) = P (X = x|X. = t) =
∏i δ
xii
γt(δ)(3)
does not dependend on θ, symm. polynomials γ. Asymptotic
theory. Likelihood ratio tests performed in the standard way.
Andersen. Asymptotic Properties of Conditional Maximum-Likelihood Estimators. Journalof the Royal Statistical Society B 1970, 32:283-301.
Andersen. The Asymptotic Distribution of Conditional Likelihood Ratio Tests. Journal ofthe American Statistical Association 1971, 66:630-3.
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Probabilistic models for measurement Copenhagen 2010
Solution of likelihood equations required recursive formulas for
the γ polynomials and for ratios
γt−x(δ(p))
γr(δ)
where δ(p) = (δi)i6=p. Procedure also yields the asymptotic co-
variance matrix. Implemented on the IBM 7094 at the Northern
Europe University Computing Center.
Andersen. The Numerical Solution of a Set of Conditional Estimation Equations. Journalof the Royal Statistical Society B 1972, 34:42-54.
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Probabilistic models for measurement Copenhagen 2010
Doctoral thesis 1973
Conditional inference as the statistical counterpart of philosoph-ical viewpoint of objective measurement.
"we must know how to relate a given model to the realworld, before alternative statistical models can berealistically discussed"
Technical rather than philosophical issues.
Rasch as official opponent took the chair addressing for a very,
very long time the interpretation of the title - should it be ”for
measuring” or ”for measurements”
Andersen. Cond. Inf. and Models for Measuring. Copenhagen: Mentalhygiejnisk Forl. 1973
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Probabilistic models for measurement Copenhagen 2010
Requirements imposed on the data by the Rasch model
(i) θ unidimensional
(ii) Pr(X1 = x1, . . . , XI |θ) =∏I
i=1 Pr(Xi = xi|θ)(iii) Pr(Xi = xi|θ, Y ) = Pr(Xi = xi|θ, Y ), for any covariate Y .
(iv) equal discrimination
(iv’)∑
i Xi sufficient for θ.
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Probabilistic models for measurement Copenhagen 2010
Rasch model implies sufficiency. The reverse is also true.
Theorem. Consider the model
f(x1, . . . , xI |θ, βi) =∏i
f(xi|θ, βi) =∏i
P (Xi = xi|θ, βi) (∗)
If T = T (X1, . . . , XI) is a minimal sufficient statistic for any value
of (β1, . . . , βI) then (*) is equivalent to the Rasch model.
Lemma. If T = T (X1, . . . , XI) is a minimal sufficient statistic for
any value of (β1, . . . , βI) then T is invariant under permutations
of X1, . . . , XI
Andersen. Conditional inference for multiple-choice questionnaires. British Journal of Math-ematical and Statistical Psychology 1973, 26:31-44.
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Probabilistic models for measurement Copenhagen 2010
Polytomous Rasch models
Algebraic generalization of Rasch, for xi = 0,1, . . . , m
Pr(Xi = xi) = c−1i exp
m∑h=1
θhx(h)i +
m∑h=1
βix(h)i
(4)
statistically treated by Andersen. Ordering properties of T .
Theorem. T = T (X1, . . . , XI) minimal suff. indep. of item para-
meters + local independence ⇒ Rasch model.
Later descriptions of polytomous Rasch model cited more often
Andersen. Sufficient statistics and latent trait models. Psychometrika 1977, 42:69-81
Andrich. A rating formul. for ordered resp. categories. Psychometrika 1978, 43:561-73.
Masters. A Rasch model for partial credit scoring. Psychometrika 1982, 47:149-74.
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Probabilistic models for measurement Copenhagen 2010
the Andersen test
Likelihood (2) works for a single score value t0 (an invariance
feature of the model). Andersen shows that the mathematics
work.
LC(δ) =( ∏
v∏
i δxvii
)/
( ∏t [γt(δ)]
nt)
where (nt) is the number of persons in each score group and
L(t0)C (δ) =
( ∏v
∏i δ
xvii
)/ [γt0(δ)]
nt0
for each t0.
Andersen. A goodness of fit test for the Rasch model. Psychometrika 1973, 38:123-40.
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Probabilistic models for measurement Copenhagen 2010
the Andersen test, II
Asympt. χ2 distributed test statistic
Z = −2 logQ = 2∑
t0 logL(t0)C (δ̂(t0))− 2 logLC(δ̂)
power considerations. Testing model against the Birnbaum model.
Graphical test.
Many, many other tests of the model followed ...
Andersen. A goodness of fit test for the Rasch model. Psychometrika 1973, 38:123-40.
Birnbaum. Some latent trait models. (In: Lord & Novick Statistical theories of mental test
scores. Reading: Addison-Wesley, 1968).
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Probabilistic models for measurement Copenhagen 2010
Glas used multinomial dist. solving problem with χ2 tests in large
contingency tables. Andersen test also overcomes this problem
Reason for lack of model studied in detail by considering residuals
r =xi|t − πi|t
s.e.(xi|t − πi|t)(5)
or
r =δ̂(t)i − δ̂i
s.e.(δ̂(t)i − δ̂i)(6)
Glas. The derivation of some tests for the Rasch model from the multinomial distribution.
Psychometrika 1988, 53:525-46.
Andersen. Residualanalysis in the polytomous Rasch model. Psychometrika 1995, 60:375-93.
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Probabilistic models for measurement Copenhagen 2010
Application of standard results yields complicated formula for the
variance V (xi|t − πi|t), but surprisingly
V (ε̂(t)i − ε̂i) = V (ε̂(t)i )− V (ε̂i) (7)
these variances being a by-product of the estimation procedure.
Andersen. Residualanalysis in the polytomous Rasch model. Psychometrika 1995, 60:375-93.
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Rao. Linear statistical inference and its applications (2nd ed.). Wiley and Sons, NY, 1973.
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Probabilistic models for measurement Copenhagen 2010
In fact (7) is a general result about variances of MLE’s
Residual diagrams to evaluate if MLE’s from independent sam-
ples can be assumed to be equal apart from random errors.
Example. Residual diagrams for the equality of the variances in
a one-way ANOVA.
Andersen. Residual Diagrams Based on a Remarkably Simple Result concerning the Vari-ances of Maximum Likelihood Estimators. Journal of Educational and Behavioral Statistics2002, 27:19-30.
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Probabilistic models for measurement Copenhagen 2010
Combining Rasch model with latent structure analysis
Rewrite likelihood based on (1) using (3):∏v
∏i Pr(Xvi = xvi|θj) = Lc(δ)
∏v g(tv|θv)︸ ︷︷ ︸
(?)
(8)
Rasch paradigm is to use first part for item analysis
"It seems, however, that the logical next step was nevertaken by Rasch, and only occasionally by his many followers"
Base inference about θ on (?). How to estimate and test assump-
tions about the latent distribution, e.g. conjugate distributions.
Andersen & Madsen. Estimating the parameters of the latent population distribution.Psychometrika 1977, 42:357-74.
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Probabilistic models for measurement Copenhagen 2010
Combining Rasch model with latent structure analysis, II
After accepting Rasch model proceed to check whether popula-
tion distribution can be described by a simple population density
Ex. 2 (Andersen & Madsen, p.370): Famed Stouffer-Toby data
Score group Item 1 Item 2 Item 3 Item 4 nt
t=0 0 0 0 0 42t=1 1 6 6 23 36t=2 7 33 33 53 63t=3 17 49 46 53 55t=4 20 20 20 20 20
Rasch model fits - normal density is clearly rejected.
Andersen & Madsen. Estimating the parameters of the latent population distribution.Psychometrika 1977, 42:357-74.
Stouffer & Toby. Role conflict and personality. Am. Journal of Sociology 1951,56:395-406.
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Probabilistic models for measurement Copenhagen 2010
Combining Rasch model with latent structure analysis, III
Testing assumptions about the structure of the underlying using
the observed score distribution also has a multivariate extension
using the contingency table
score at time 1 × score at time 2
Latent regression models imposing a linear structure on θ
Andersen. Est. latent corr. between repeated testings. Psychometrika 1985, 50:3-16.
Andersen. Latent Regression Analysis. Research Report 106, Department of Statistics,University of Copenhagen, 1994.
Andersen. Latent Regression Analysis based on the Rating Scale Model. PsychologyScience 2004, 46:209-26.
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Probabilistic models for measurement Copenhagen 2010
The Rasch model
MATHEMATICS
ξδi1+ξδi
STATISTICS PHILOSOPHY
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Probabilistic models for measurement Copenhagen 2010
Georg Rasch
MATHEMATICS↘
ξδi1+ξδi
↘
STATISTICS PHILOSOPHY
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Probabilistic models for measurement Copenhagen 2010
Erling Andersen
MATHEMATICS
ξδi1+ξδi
↙↗
STATISTICS PHILOSOPHY
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Probabilistic models for measurement Copenhagen 2010
Erling B. Andersen
Developed a lot of theory for the Rasch model, but also did a
lot of applied work using the Rasch model. Mainly at the Danish
Research Institute for Mental Health.
Placed the Rasch model within mainstream statistics.
Used the Rasch model as a platform to derive general results.
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Probabilistic models for measurement Copenhagen 2010
Impact of Erling B. Andersens work
first proof that Rasch models are characterized by sufficiency
proof that conditional ML estimates are consistent
theory of conditional ML estimation
first to combine Rasch models and latent structure analysis.
1995 Rasch book:
references in 16 out of 21 chapters - 39 in total (disregarding
cases where authors refer to themselves Erling B. Andersen is
the person with the largest number of citations).
Fischer & Molenaar. Rasch models. Foundations, recent developments, and applications.New York: Springer-Verlag, 1995.
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Probabilistic models for measurement Copenhagen 2010
Philosophy vs. mathematical statistics
Idea of conditional inference came from Rasch. The theory was
developed by Erling B. Andersen
Erling the model as a statistical model and never subscribed to
the idea that it should be regarded as a special ’measurement’.
Never referred to the concept of specific objectivity in any of his
papers.
Practice of Rasch analysis today owes more to Erling B. Andersen
than it does to Rasch.
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