Volkert Siersma Research Unit for General Practice in Copenhagen [email protected] P γ...

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Volkert Siersma Research Unit for General Practice in Copenhagen http://www.gpract.ku.dk [email protected] P γ measure for association between categorical variables with partial or tentative ordering of categories
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Page 1: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

Volkert Siersma

Research Unit for General Practice in Copenhagen

http://www.gpract.ku.dk [email protected]

Pγ measure for association between categorical variables with partial or

tentative ordering of categories

Page 2: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

Categorical variables

Nominal variables Ordinal variables

Ordinal variables are categorical variables with additional information in the ordering of the categories.

This can be used to devise stronger and more meaningful analyses between ordinal variables.

Categories have an inherent ordering

Categories are unordered relative to each other

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Page 3: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

Categorical inference

Nominal variables Ordinal variables

Tests for conditional independence in multidimensional contingency tables

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Page 4: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

Categorical inference

Nominal variables Ordinal variables

Inference based on χ2-measures.

LR test with saturated alternative

LR test with 2-factor alternative

Tests for conditional independence in multidimensional contingency tables

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Page 5: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

Categorical inference

Nominal variables Ordinal variables

Inference based on χ2-measures.

LR test with saturated alternative

LR test with 2-factor alternative

Inference based on χ2-measures.

Tests for conditional independence in multidimensional contingency tables

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Page 6: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

Categorical inference

Nominal variables Ordinal variables

Inference based on χ2-measures.

LR test with saturated alternative

LR test with 2-factor alternative

Inference based on χ2-measures.

Inference based on rank correlation measures.

Tests for conditional independence in multidimensional contingency tables

Goodman and Kruskal’s γ measure

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Page 7: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

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Self-rated health

Smoking Excellent Good Fair Poor Very poor

Never 24 84 95 18 2

Previous 29 77 102 20 3

Current 31 82 121 18 4

An example

Type 2 diabetes patients at diagnosis

Page 8: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

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Self-rated health

Smoking Excellent Good Fair Poor Very poor

Never 3.4% 11.8% 13.4% 2.5% 0.3%

Previous 4.1% 10.9% 14.4% 2.8% 0.4%

Current 4.4% 11.6% 17.0% 2.5% 0.6%

An example

…the corresponding empirical probability table

Page 9: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

3An example

If the two variables are independent then the joint probability function, i.e. the cell probabilities of the table, are just the products of the marginal probabilities of the categories of each of the variables

P(X=x and Y=y) = P(X=x)*P(Y=y)

Page 10: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

3An example

…the marginal probability distributions

Self-rated health

Excellent Good Fair Poor Very poor

Smoking 11.8% 34.2% 44.8% 7.9% 1.3%

Never 31.4%

Previous 32.5%

Current 36.1%

Page 11: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

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Self-rated health

Smoking Excellent Good Fair Poor Very poor

Never 24/26 84/76 95/100 18/18 2/3

Previous 29/27 77/79 102/103 20/18 3/3

Current 31/30 82/88 121/115 18/20 4/3

An example

…observed versus expected

Page 12: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

3An example

Compare the observed table and the table expected under independence.

Pearson’s statistic, which is based on the sum of squared differences between the observed and the expected table entries, is chi-squared distributed when the null hypothesis is true.

Here: df=8 and p=0.94

Or we perform an exact test!

Page 13: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

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Self-rated health

Smoking Excellent Good Fair Poor Very poor

Never 24 84 95 18 2

Previous 29 77 102 20 3

Current 31 82 121 18 4

An example

Goodman and Kruskal’s γ.

Two independent draws (X1,Y1) and (X2,Y2) from the joint (X,Y) distribution.

Page 14: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

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Self-rated health

Smoking Excellent Good Fair Poor Very poor

Never 24 84 95 18 2

Previous 29 77 102 20 3

Current 31 82 121 18 4

An example

Two independent draws (X1,Y1) and (X2,Y2) from the joint (X,Y) distribution.

Concordance ”If X goes up, Y goes up.”

X1

X2

Y1 Y2

Page 15: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

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Self-rated health

Smoking Excellent Good Fair Poor Very poor

Never 24 84 95 18 2

Previous 29 77 102 20 3

Current 31 82 121 18 4

An example

X1

X2

Y1Y2

Concordance ”If X goes up, Y goes up; if X goes down, Y goes down” or ”X and Y move in the same direction.”

The definition is symmetric

Page 16: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

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Self-rated health

Smoking Excellent Good Fair Poor Very poor

Never 24 84 95 18 2

Previous 29 77 102 20 3

Current 31 82 121 18 4

An exampleDiscordance ”If X goes up, Y goes down.” or ”X and Y move in opposite directions.”

X1

X2

Y1 Y2

Page 17: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

3An example

nce)P(discordance)P(concorda

nce)P(discordance)P(concordaγ

Goodman and Kruskal’s γ. Difference of the probabilities for concordance and discordance scaled with the probability of not having ties.

Here: γ=0.02 and p=0.60

Page 18: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

Partial order

Nominal Ordinal

Only part of the categories is ordered.

Goals in a weight control programme:

No goal set Keep current weight Reduction < 2 kg Reduction < 4 kg Reduction < 6 kg Reduction > 6 kg

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Page 19: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

Partial order

Nominal Ordinal

Only part of the categories is ordered.

Goals in a weight control programme:

No goal set Keep current weight Reduction < 2 kg Reduction < 4 kg Reduction < 6 kg Reduction > 6 kg

Extra-ordinal category

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Page 20: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

Partial order

Nominal Ordinal

Only part of the categories is ordered.

Goals in a weight control programme:

No goal set Keep current weight Reduction < 2 kg Reduction < 4 kg Reduction < 6 kg Reduction > 6 kg

Has to be treated as nominal variable and the information in the ordering is lost.

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Page 21: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

Partial order

Nominal Ordinal

Only part of the categories is ordered.

Goals in a weight control programme:

No goal set Keep current weight Reduction < 2 kg Reduction < 4 kg Reduction < 6 kg Reduction > 6 kg

No indication on the effect of the extra-ordinal category in relation to the others.

?

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Page 22: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

Nominal Ordinal

Tentative order

The ordering of the categories is of interest.

Danish political parties:

Ø SF A B Q CD Z V C DF

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Page 23: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

Nominal Ordinal

Tentative order

The ordering of the categories is of interest.

Danish political parties:

Ø SF A B Q CD Z V C DF

Ordering w.r.t. left-right affiliation

Methods for nominal variables do not give information on the nature of the relationship.

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Page 24: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

Ordinal information

Partially ordinal variables:

have to be treated as nominal variables in general

information in the ordering of the categories, and statistical power, is lost.

Tentatively ordinal variables:

the form of the association has to be deducted by examination of stratified tables or parameters of loglinear models

which in multivariate analysis can be most confusing.

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Page 25: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

An ordering X(r) of X is an ordinal random variable with a specific permutation r of the categories of X.

If X has a (partial) order, we regard only valid orderings of X, i.e. orderings based on permutations that do not violate this partial order.

Nominal variable: all orderings are valid

Ordinal variable: only one ordering is valid

An ordering of a categorical variable7

Page 26: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

A Pγ measure of association

The Pγ measure of association between a partially ordered or nominal X and an ordinal Y: the maximum γ between a valid ordering of X and Y.

X(r)Yr

XYP γmaxγ

The optimal monotone ordering of X w.r.t. Y: the valid ordering of X for which this maximum is obtained.

X(r)Yr

opt γargmaxr

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Page 27: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

A partial γ measure of association

In multidimensional contingency tables one is often interested in the relationship of two variables, X and Y, conditional on (controlled for, stratified by) a third variable Z.

Within each stratum of Z, a γ measure is calculated between X and Y.

A partial γ measure of monotone association between X and Y is defined as a weighted summary γ measure across subtables spanned by the categories of Z.

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Page 28: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

The partial PγXY|Z between a partially ordered or nominal X and an ordinal Y conditional on a nominal Z is defined as the maximum partial γ between a valid ordering of X and Y.

The partial optimal monotone ordering of X w.r.t. Y, controlled for Z is the ordering corresponding to the partial PγXY|Z.

A partial Pγ measure of association10

Page 29: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

Inference

Significance of the Pγ measure and its corresponding partial measure is assessed by comparison of the obtained value with a simulated distribution under the null hypothesis where X and Y are independent.

Resampling tests are standard in the analysis of multi-way contingency tables as tests based on the asymptotic distribution are of very low power

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Page 30: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

Simulation study

Relationship between X and Y conditional on Z

X and Y ordinal, Z nominal.

Dim(X) = 3 or 5

Dim(Y) = 3 or 5

Dim(Z) = 2 or 10

Uniform marginal distributions

N = 200

partial γ = 0 or 0.15

Categories of Y are permuted to calculate Pγ

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Page 31: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

Simulation study – results γ=0

The attained level of significance, i.e. the power of the tests when the true γ is 0, has to be 5%.

Our results show that this is not a problem. All MC estimates of the critical value are in the 95% confidence region:

0.05 0.0135

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Page 32: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

Simulation study – results γ=0.15

# rows

# columns

# strata

(XZ,YZ) against (XYZ)

(XZ,YZ) against

(XY,XZ,YZ)

Partial γ Partial Pγ

Correct ordering

Correlation with true ordering (mean)

3 3 2 0.178 0.229 0.377 0.283 59.6% 0.70

3 3 10 0.084 0.221 0.343 0.280 58.2% 0.68

3 5 2 0.137 0.193 0.433 0.242 8.6% 0.54

3 5 10 0.074 0.202 0.425 0.255 8.9% 0.53

5 3 2 0.158 0.247 0.431 0.359 61.8% 0.72

5 3 10 0.079 0.196 0.408 0.329 62.0% 0.70

5 5 2 0.158 0.231 0.553 0.347 12.0% 0.60

5 5 10 0.097 0.219 0.518 0.312 10.8% 0.57

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Page 33: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

Simulation study – results γ=0.15

# rows

# columns

# strata

(XZ,YZ) against (XYZ)

(XZ,YZ) against

(XY,XZ,YZ)

Partial γ Partial Pγ

Correct ordering

Correlation with true ordering (mean)

3 3 2 0.178 0.229 0.377 0.283 59.6% 0.70

3 3 10 0.084 0.221 0.343 0.280 58.2% 0.68

3 5 2 0.137 0.193 0.433 0.242 8.6% 0.54

3 5 10 0.074 0.202 0.425 0.255 8.9% 0.53

5 3 2 0.158 0.247 0.431 0.359 61.8% 0.72

5 3 10 0.079 0.196 0.408 0.329 62.0% 0.70

5 5 2 0.158 0.231 0.553 0.347 12.0% 0.60

5 5 10 0.097 0.219 0.518 0.312 10.8% 0.57

Considerably higher power than the other tests.

This was to be expected because the data was generated with a monotone relationship.

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Page 34: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

Simulation study – results γ=0.15

# rows

# columns

# strata

(XZ,YZ) against (XYZ)

(XZ,YZ) against

(XY,XZ,YZ)

Partial γ Partial Pγ

Correct ordering

Correlation with true ordering (mean)

3 3 2 0.178 0.229 0.377 0.283 59.6% 0.70

3 3 10 0.084 0.221 0.343 0.280 58.2% 0.68

3 5 2 0.137 0.193 0.433 0.242 8.6% 0.54

3 5 10 0.074 0.202 0.425 0.255 8.9% 0.53

5 3 2 0.158 0.247 0.431 0.359 61.8% 0.72

5 3 10 0.079 0.196 0.408 0.329 62.0% 0.70

5 5 2 0.158 0.231 0.553 0.347 12.0% 0.60

5 5 10 0.097 0.219 0.518 0.312 10.8% 0.57

The test based on P is not as powerful as the one based on .

The power is higher than both the LR tests considered here.

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Page 35: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

Simulation study – results γ=0.15

# rows

# columns

# strata

(XZ,YZ) against (XYZ)

(XZ,YZ) against

(XY,XZ,YZ)

Partial γ Partial Pγ

Correct ordering

Correlation with true ordering (mean)

3 3 2 0.178 0.229 0.377 0.283 59.6% 0.70

3 3 10 0.084 0.221 0.343 0.280 58.2% 0.68

3 5 2 0.137 0.193 0.433 0.242 8.6% 0.54

3 5 10 0.074 0.202 0.425 0.255 8.9% 0.53

5 3 2 0.158 0.247 0.431 0.359 61.8% 0.72

5 3 10 0.079 0.196 0.408 0.329 62.0% 0.70

5 5 2 0.158 0.231 0.553 0.347 12.0% 0.60

5 5 10 0.097 0.219 0.518 0.312 10.8% 0.57

The influence of the simulation parameters is intuitive.

This becomes clear in more extensive simulations.

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Page 36: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

Simulation study – results γ=0.15

# rows

# columns

# strata

(XZ,YZ) against (XYZ)

(XZ,YZ) against

(XY,XZ,YZ)

Partial γ Partial Pγ

Correct ordering

Correlation with true ordering (mean)

3 3 2 0.178 0.229 0.377 0.283 59.6% 0.70

3 3 10 0.084 0.221 0.343 0.280 58.2% 0.68

3 5 2 0.137 0.193 0.433 0.242 8.6% 0.54

3 5 10 0.074 0.202 0.425 0.255 8.9% 0.53

5 3 2 0.158 0.247 0.431 0.359 61.8% 0.72

5 3 10 0.079 0.196 0.408 0.329 62.0% 0.70

5 5 2 0.158 0.231 0.553 0.347 12.0% 0.60

5 5 10 0.097 0.219 0.518 0.312 10.8% 0.57

Insight is gained in the ordering of the categories.

The identification of the correct ordering depends on the number of categories that is permuted.

The ordering will be close to, but unlikely to be the correct ordering.

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Page 37: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

The relation between γ and Pγ

Dim(X) = 5

Dim(Y) = 5

Dim(Z) = 10

γ = 0.15

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Page 38: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

The relation between γ and Pγ

|γ|γP

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Page 39: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

The relation between γ and Pγ

|γ| is closer to Pγ when the estimated values for these coefficients are higher

|γ|γP

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Page 40: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

The distribution of Pγ

γ = 0

Normal?!?

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Page 41: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

The distribution of Pγ

γ = 0.15

Normal…

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Page 42: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

Danish political parties

European Values Studies

Denmark:

survey in 1981, 1990 and 1999

preferred political party (10 parties)

political attitudes measured on a left-right discrete (10 point) VAS scale

10 x 10 x 3 table

Significance of the assiciation is obvious

Ordering of the parties is common knowledge (up to a certain level…)

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Page 43: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

Danish political parties - Pγ

Left Right categories |Far left |The Red-Green Alliance Ø |The Socialist People’s party SF |The Social Democratic Party A |The Social Liberal Party B |The Christian People’s party Q |The Centre Democrats CD |The Progress Party Z |The Liberal Party V |The Conservative Party C |The Danish People’s Party DF | | |Far right

P = 0.629 a very strong association

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Page 44: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

Danish political parties - Pγ

Left Right categories |Far left |The Red-Green Alliance Ø |The Socialist People’s party SF |The Social Democratic Party A |The Social Liberal Party B |The Christian People’s party Q |The Centre Democrats CD |The Progress Party Z |The Liberal Party V |The Conservative Party C |The Danish People’s Party DF | | |Far right

Common knowledge:

Left (in this order)

Center

Right

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Page 45: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

Danish political parties - Pγ

Left Right categories |Far left |The Red-Green Alliance Ø |The Socialist People’s party SF |The Social Democratic Party A |The Social Liberal Party B |The Christian People’s party Q |The Centre Democrats CD |The Progress Party Z |The Liberal Party V |The Conservative Party C |The Danish People’s Party DF | | |Far right

The position of DF on the far right is somewhat surprising

New party in 1999

Reflects the political attitudes of the persons preferring DF to other parties in 1999

Since then, the party has with some success attempted to move towards the middle of the spectrum

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Page 46: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

A weight control program19

Weight goals against attained weight.

A considerable number have no goal set.

Page 47: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

A weight control program19

For convenience we code the categories of the weight goal variable with letters.

Page 48: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

A weight control program19

We investigate the placement of the no goal set category with Pγ.

The relationship is significant, but confounded.

Page 49: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

A weight control program19

We investigate the no goal set category with the partial Pγ.

Page 50: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

A weight control program19

We investigate the no goal set category with the partial Pγ conditional on many possible confounders.

Page 51: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

A weight control program19

We construct the product of weight goal and weight at current session. The optimal monotone ordering with respect to attained weight shows effect modification.

Page 52: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

A weight control program19

For normal or extremely obese people no goal set is associated with low attained weight, but others are better off with guidance in the form of weight goals.

Page 53: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

Metabolic syndrome

Combinations (products) of ordinal variables

sex x age → C

hyperglycaemia x obesity → D

Metabolic Syndrome

No Yes D

sex age No Yes No Yes C C 1 4 3 2

M Y 45 15 14 5 1 3 83 9 5 2

M O 12 9 18 12 2 4 25 11 12 0

F Y 83 2 5 9 3 1 45 5 14 15

F O 25 0 12 11 4 2 12 12 18 9

D 1 2 3 4 Pγ = 0,53

Hyperglycaemia

Obesity

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Page 54: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

Metabolic syndrome

Categories of C and D permuted to obtain Pγ

• omnibus tests

• screening priority

• scale construction

Metabolic Syndrome

No Yes D

sex age No Yes No Yes C C 1 4 3 2

M Y 45 15 14 5 1 3 83 9 5 2

M O 12 9 18 12 2 4 25 11 12 0

F Y 83 2 5 9 3 1 45 5 14 15

F O 25 0 12 11 4 2 12 12 18 9

D 1 2 3 4 Pγ = 0,53

Hyp.glyc.

Obesity

D 1 2 3 4

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Page 55: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

Metabolic syndrome

It is difficult to interpret the obtained orderings as something clinically meaningful.

Metabolic Syndrome

No Yes D

sex age No Yes No Yes C C 1 4 3 2

M Y 45 15 14 5 1 3 83 9 5 2

M O 12 9 18 12 2 4 25 11 12 0

F Y 83 2 5 9 3 1 45 5 14 15

F O 25 0 12 11 4 2 12 12 18 9

D 1 2 3 4 Pγ = 0,53

Hyp.glyc.

Obesity

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Page 56: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

Canonical correlations

Can also find the optimal monotone ordering of one nominal variable relative to one ordinal variable. The extension to a partial analysis is troublesome.

Can also be used to construct omnibus tests. But these do not take the ordinal nature of the variables into account.

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Page 57: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.

Conclusion

Pγ based inference is more powerful than nominal χ2 based inference.

Pγ based inference makes use of information in a partial ordering of the categories.

Pγ based inference gives information on the ordering of the categories.

Pγ based inference gives insight in effect modification.

Pγ based inference allows for omnibus tests.

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Page 58: Volkert Siersma Research Unit for General Practice in Copenhagen  V.Siersma@gpract.ku.dk P γ measure for association between categorical.