Introduction - the difference to the linear involution

There are many research fields with those the dependent variable a discrete development has (a certain event steps or not; e.g. illness, accident, etc.). In a linear regression analysis becomes - as the name already says - a linear relationship between the interesting and the explaining variables computes. The goal here is to find the process of the looked for straight line (and/or a surface or a more highly dimensional surface with a multivariate involution) which approximates the empirical point distribution as well as possible. The problem thereby is that a functional connection between a binary Response variable Y [0; 1] and a factor of influence X in direct way, i.e. in form of a linear connection to be not meaningfully specified knows, there in the linear involution model the dependent variable metrically scaled is theoretical and all developments of - 8 to +8 to realize can (see also scale level).

In addition with "the usual "involution not the probability for the occurrence of a certain development by Y, but concrete development is predicted by Y. The influences on such variables cannot be examined thus with the procedure of the linear regression analysis, since substantial conditions for application are not given in inference-doistic regard (normal distribution of the residues, variance homogeneity) in particular. Furthermore a linear involution model can lead with such variables to inadmissible forecasts: If one codes two developments of the dependent variables with 0 and 1, then one can understand the forecast of a linear involution model as forecast of the probability that the dependent variable takes the value 1 - formally: P (Y=1) -, but can come it to the fact that values are predicted outside of this range.

Advantages of the logistic involution

The logistic regression analysis offers an elegant possibility of examining the influence several (of quantitative or qualitative) arguments or "factors of risk "for a dichotome goal size - nonlinear -. The idea of the logistic involution is based on the conception that the probability of an event with an involution model can be functionally described. The influence of the arguments can be modelled so directly. A further advantage is that this variables can be usually transferred in their original form to the model. Besides only the involution coefficients must become estimated, which reduces the number of necessary statistic tests.

The logistic involution solves this problem by a suitable transformation of the dependent variable P (Y=1). It goes out from the idea of the Odds, i.e. the relationship of P (Y=1) to the Gegenwahrscheinlichkeit 1-P (Y=1) and/or P (Y=0) (on coding the alternative category with 0). In principle it concerns a nonlinear transformation of the linear involution with a range of values between 0 and 1 (contrary to a standard normal distribution and/or a per bit distribution).

\ mathrm {Odds} (Y_ {1/0}) = \ frac {\ mathrm {P} (Y=1)}{1 \ mathrm {P} (Y=1)}= \ frac {\ mathrm {P} (Y=1)}{\ mathrm {P} (Y=0)}

The Odds can take values more largely 1, but is downward limited their range of values (it approaches iterated 0 on). Unrestricted range of values becomes by the transformation of the Odds into the so-called Logits

\ mathrm {Logit} (Y_ {1/0}) = \ LN \ frac {\ mathrm {P} (Y=1)}{1 \ mathrm {P} (Y=1)}

obtained; these can take values plus infinitely between minus and. A Odds reason (OR) expresses thus the effect of an argument on the dependent variable. The OR represents the factor, by which per unit of the arguments the probability of the dependent variables reduces increased/.

In the logistic involution then the regression equation becomes

\ mathrm {Logit} (Y_ {1/0} |X) =b_0+b_1X_1+ \ dots+b_kX_k

estimated; involution weights are thus determined, after which the estimated Logits for a given matrix can be computed by arguments X. Following graphics shows, how Logits (x axis) with the output probabilities P (Y=1) (y axis) is connected:

The involution coefficients of the logistic involution are not simple to interpret. Therefore one forms frequently the so-called effect coefficients by formation of the antilogarithm; the regression equation refers thereby to the Odds:

\ mathrm {Odds} (Y_ {1/0} |X) = \ exp (b_0+b_1X_1+ \ dots+b_kX_k)

The coefficients exp (bn) are called often also effect coefficients. Here coefficients designate smaller 1 a negative influence on the Odds, a positive influence are given, if exp (bn) > 1.

By a further transformation the influences of the logistic involution can be expressed also than influences on the probabilities P (Y=1):

\ mathrm {P} (Y=1|X) = \ frac {\ exp (b_0+b_1X_1+ \ dots+b_kX_k)}{1+ \ exp (b_0+b_1X_1+ \ dots+b_kX_k)}

The involution parameters become estimated on the basis maximum of the Likelihood procedure. Inference-statistic procedures are available both for the individual involution coefficients and for the universal model (see forest statistics and Likelihood relationship test); in analogy to the linear involution model also procedures of the involution diagnostics were developed, on the basis those individual cases with extra large influence on the result of the model estimation to be identified to be able. Finally there are also some suggestions on the computation of a size, which explained in analogy to the R2 of the linear involution an estimation "to variance" permitted; one speaks here of so-called Pseudo-R2.

A transmission of the logistic involution (and the pro bit model) on dependent variable with more than two (nominal or ordinalskalierten) characteristics is possible (see Multinomiales Logit and Ordinales Logit.)

Literature

• Agresti, Alan: Categorical DATA analysis. New York: Wiley, 1990.
• Andress, Hans J. - A. Steffen: Analysis of tables and data kategorialen. Berlin and others: Springer, 1997.
• Hosmer, D.W. and S. Lemeshow: Applied logistic involution. New York; Chichester, Wiley, 2000.
• Small tree, D.G., L.L. Kupper and K.E. Muller: Applied involution analysis and OTHER multivariable Methods. Boston, PWS Kent, 1987.

Articles in category "Logistic involution"

We found here 1 articles.

Related Websites

We found here 3 related websites.

• From Involution to Development: Household Risk Taking and Market ...
  However, peasants


• PSIONICS: 11/01/2002 - 11/30/2002
  ... fissure expansion and manipulation, logistic involution, breakage, obviated attacks, secret transparencies magnifiying micrometriculated chasms, ...


• RETINOPATHY OF PREMATURITY: INVOLUTION, FACTORS PREDISPOSING TO ...
  To characterize involution of retinopathy of prematurity (ROP) following ... Univariate and multivariate logistic regression models were constructed to ...