Standard errors and z- or t-statistics for binomial models in .NET Encoding PDF417 in .NET Standard errors and z- or t-statistics for binomial models

8.5.2 Standard errors and z- or t-statistics for binomial models using none tointegrate none on asp.net web,windows application CBC The z- or t-s none for none tatistics are sometimes known as the Wald statistics. They are based on an asymptotic approximation to the standard error. For binomial models, this approximation can be seriously inaccurate when the tted proportions are close to 0 or 1, to the extent that an increased difference in an estimated proportion can be associated with a smaller t-statistic.

Where it is important to have a reasonably accurate p-value, this is suitably obtained from an analysis of deviance table that compares models with and without the term.. Generalized linear models and survival analysis 0.05 0.5 0.

95. Leverage logit link probit link cloglog link 0.01 0.2 0.8 0.99 Fitted proportion Figure 8.12 L none none everage versus tted proportion, for the three common link functions. Points that all have the same binomial totals were symmetrically placed, on the scale of the response, about a tted proportion of 0.

5. The number and location of points will affect the magnitudes, but for the symmetric links any symmetric con guration of points will give the same pattern of change..

The following demonstrates the effect:. fac <- fac none for none tor(LETTERS[1:4]) p <- c(103, 30, 11, 3)/500 n <- rep(500,4) summary(glm(p fac, family=binomial, Estimate Std. Error z value (Intercept) -1.35 0.

111 -12.20 facB -1.40 0.

218 -6.42 facC -2.45 0.

324 -7.54 facD -3.76 0.

590 -6.38 > > > >. weights=n))$coef Pr(> z. ) 3.06e-34 1. 35e-10 4.

71e-14 1.78e-10. Notice that t none for none he z value for level D is smaller than the z value for level C, even though the difference from level A (the reference) is greater. The phenomenon is discussed in Hauck and Donner (1977)..

8.5.3 Leverage for binomial models In an lm mode none none l with a single predictor and with homogeneous errors, the points that are furthest away from the mean inevitably have the largest leverages. For models with binomial errors, this is not the case. The leverages are functions of the tted values.

Figure 8.12 shows the pattern of change of the leverage, for the three common link functions, when points are symmetrically placed, on the scale of the response, about a tted proportion of 0.5.

The optimal placement of points for a probit link is slightly further apart than for a logit link.. 8.6 Models wi th an ordered categorical or categorical response This section draws attention to an important class of models that further extend the generalized linear model framework. Ordinal regression models will be discussed in modest detail.

. 8.6 Models with an ordered categorical or categorical response Table 8.2 Dat a from a randomized trial assessments of the clarity of the instructions provided to the inhaler..

Easy (categor none for none y 1) Inhaler 1 Inhaler 2 99 76 Needed rereading (category 2) 41 55 Not clear (category 3) 2 13. Table 8.3 Odd s ratios, and logarithms of odds ratios, for two alternative choices of cutpoint in Table 8.2.

. Easy versus s none none ome degree of dif culty Odds Inhaler 1 Inhaler 2 99/43 76/68 Log odds log(99/43) = 0.83 log(76/68) = 0.11 Clear after study versus not clear Odds 140/2 131/13 Log odds log(140/2) = 4.

25 log(131/13) = 2.31. Loglinear mod els, which may be appropriate when there is a qualitative (i.e., unordered) categorical response, will get brief mention.

. 8.6.1 Ordinal regression models We will demon strate how logistic and related generalized linear models that assume binomial errors can be used for an initial analysis. The particular form of logistic regression that we will demonstrate is proportional odds logistic regression. Ordinal logistic regression is relevant when there are three or more ordered outcome categories that might, e.

g., be (1) complete recovery, (2) continuing illness, (3) death. Here, in Table 8.

2, we give an example where patients who were randomly assigned to two different inhalers were asked to compare the clarity of lea et instructions for their inhaler (data, initially published with the permission of 3M Health Care Ltd, are adapted from Ezzet and Whitehead, 1991).. Exploratory analysis There are two ways to split the outcomes into two categories: we can contrast easy with the remaining two responses (some degree of dif culty), or we can contrast the rst two categories (clear, perhaps after study) with not clear . Table 8.3 presents, side by side in parallel columns, odds based on these two splits.

Values for log(odds) are given alongside. Wherever we make the cut, the comparison favors the instructions for inhaler 1. The picture is not as simple as we might have liked.

The log(odds ratio), i.e., the difference on the log(odds) scale, may depend on which cutpoint we choose.

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