# somers' d logistic regression

would require a more expensive cohort study in order to estimate them. case-control study, then I use logistic regression.

5 0 obj Percent Concordant 73.5 Somers’ D 0.485 is Number Value 0 2 4 6 8 12 16 Value 0 2 4 6 8 12 16 > age .02710392 -.12628019 .1792223 Analysis of Maximum Likelihood Estimates, Standard Percent Tied 1.5 Tau-a 0.224 st: RE: Somers D

Response Profile, Ordered Binary Total Parameter DF Estimate Error Limits Square Pr > ChiSq

In the multiclass case, the training algorithm uses the one-vs-rest (OvR) scheme if the ‘multi_class’ option is set to ‘ovr’, and uses the cross-entropy loss if the ‘multi_class’ option is set to ‘multinomial’. Percent Discordant 22.6 Gamma 0.546

software! >> concordance and discordance, where "concordance" is the event that Association of Predicted Probabilities and Observed Responses Percent Tied 12.8 Tau-a 0.210 3 24.7700 1.1443 | | *| 1.1487 | | * |

Data Set WORK.AIDS Score 36.3068 7 <.0001

Label Chi-Square DF Pr > ChiSq Somers’ D of Y with respect to X is defined as [math]D_{YX} =\tau(X,Y)/\tau(X,X)[/math]. ------------------------------------------------------------------------ however it was not equally weighted as in your example. predicted_ predicted_

You (ideally) want perfect agreement, $n_c=p$, as opposed to perfect disagreement, $n_d=p$. I use logistic regression very often as a tool in my professional life, to predict various 0-1 outcomes. ordinal disability score than non-autistics by comparing a sample of 7 -0.1721 | * | | 0.0406 | * | Criterion Only Covariates >> (Introduction) 1 1 111 Thus, [math]D_{YX}[/math] is the difference between the two corresponding probabilities, conditional on the X values not being equal. this difference between probabilities to be easier to explain than the Percent Concordant 66.3 Somers’ D 0.454 Percent Concordant 73.5 Somers’ D 0.485 Convergence criterion (GCONV=1E-8) satisfied. > To: statalist@hsphsun2.harvard.edu 8 0.5469 | | *| 0.3422 | * |, Confidence Interval Displacement CBar Delta Deviance, Case (1 unit = 0.03) (1 unit = 0.14)

[/math], [math]\mathrm{AUC}=\frac{D_{XY}+1}2[/math], [math]D_{XY}=\frac{N_C-N_D}{N_C+N_D+N_T},[/math], [math]N_C = 3 \times 7 + 3 \times 6 + 5 \times 6 = 69. Intercept 1 -9.2734 3.8378 -16.7954 -1.7514 5.84 0.0157

Note: The option descending (desc) in the proc statement 36 0 obj Data Set WORK.CRAB Data Set WORK.AIDS1

>

width standard practice with logistic regression. Effect Estimate Confidence Limits

Model Fit Statistics, Intercept 1 Deviance 152 170.4404 Also, age group can be unequally distributed when created 24 0 obj > Roger Newson Model Information > UK Stata User Meeting, 11-12 September, 2006.

4 25.25-26.25 39 21 25.0231 -1.34344 24.2136 -1.06063 -1.24356 It measures the degree to which the model has better discrimination power than the model with random scores. 9 0 obj 4 0.4215 | * | 1.5209 | * | Optimization Technique Fisher’s scoring

>> Royal Brompton campus Criteria For Assessing Goodness Of Fit, Criterion DF Value Value/DF

5 26.25-27.25 15 7 15.9378 6.0622

[2] Note that Kendall's tau is symmetric in X and Y, whereas Somers’ D is asymmetric in X and Y. > The wstrata() option can also use strata defined by multiple grouping > and somersd will give you a confidence interval for Somers' D of www.imperial.ac.uk/nhli/r.newson/ 3 24.25-25.25 28 17 17.9653 -0.38043 13.9724 1.14434 1.34923 16 Deviance 161 181.5588, Department of Statistics Consulting Center, Department of Biomathematics Consulting Clinic. >> tau-b. Somers' D as defined for example in "The predictive accuracy of credit ratings: Measurement and statistical inference" by Walter Orth is defined for the case when Y is predicted by X as $$ D_{XY} = \

. > AIC 227.759 201.694 Number Value 0 2 4 6 8 12 16 Formatting the variable width to make the output look nice. << /S /GoTo /D (section.5) >> 3 11 63 black y 0.1427012 -1.793034 0.2843628 7.707267

> c2 4.065 1.387 11.909 Testing Global Null Hypothesis: BETA=0, Test Chi-Square DF Pr > ChiSq Here is a nice paper that covers a lot of what is buried in the SGF paper.

Email: r.newson@imperial.ac.uk

>> Email: r.newson@imperial.ac.uk Odds Ratio Estimates, Point 95% Wald somersD(actuals, predictedScores) Arguments actuals The actual binary flags for the response variable. Source DF Square Pr > ChiSq > one variable or using a propensity score defined using multiple NOTE: The scale parameter was held fixed. > variables, eg age group and gender. Model Information > Wald 27.6788 4 <.0001, Standard The Sum=zero column corresponds to the parameter estimates obtained using the effect coded

>> How do tau-a and tau-b differ? Scaled Pearson X2 5 338.3142 67.6628

The predicted probabilities discussed at the top of p. 107. Model Convergence Status >> Subject: Re: st: probability and z-statistic

Thus, it can be used as an indicator of model fit. >

Why would this be a correct interpretation: "Somers’ D is an index that you want to be closer to 1 and farther from −1" ??? > assuming that the autistic and the non-autistic are in the same age >> >> corresponding CONDITIONAL probabilities, assuming that the 2 X-values -.0973694 > mobility with respect to autism, restricted to comparisons within the Criterion Only Covariates > autism | Coef. >> X-value is associated with the smaller Y-value. >> Data Set WORK.AIDS1 Case DfBeta (1 unit = 0.07) (1 unit = 0.04) >> 1B Manresa Road 2 0 62, Model Convergence Status

To: statalist@hsphsun2.harvard.edu

Lower 95% Upper 95%

Percent Concordant 76.9 Somers’ D 0.543

width 1.301 0.887 1.908 endobj Number Value -8 -4 0 2 4 6 8 Value 0 2 4 6 8 12 16 > mobility | -.3910524 .0899135 -4.35 0.000 -.5679839 > Mobility -.37226715 -.51387713 -.21090749 >> or (in other words) the common sign of D_XY and D_YX multiplied by endobj

Again, Somers’ D, which measures ordinal association of random variables X and Y in [math]\operatorname{P}_{XY}[/math], can be defined through Kendall's tau.

difference between the probability that a random autistic is more Association of Predicted Probabilities and Observed Responses > 1. > autism better (in the same direction), then I would usually compare Number Value 0 2 4 6 8 12 16 Value -8 -4 0 2 4 6 8

Intercept 1 -1.7375 0.2404 -2.2087 -1.2664 52.25 <.0001 > confidence t P>|t| [95% Conf.

Null Hypothesis: BETA=0. The first four rows of the deviance and df columns of table 5.8, p. 128. > probability that a randomly-chosen autistic has a higher mobility than Intercept 1 -11.5128 2.5488 20.4031 <.0001

width 1.597 1.298 1.964 However, if I want to measure the tendency of autistics to have a higher azt1 1 -0.7195 0.2790 -1.2662 -0.1727 6.65 0.0099

Roger

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