lsm (version 0.2.0)

lsm: Estimation of the log Likelihood of the Saturated Model

Description

When the values of the outcome variable Y are either 0 or 1, the function lsm() calculates, among others, the values of the maximum likelihood estimates (ML-estimations) of the corresponding parameters in the null, complete, saturated and logistic models and also the estimations of the log likelihood in each of this models. The models null and complete are described by Llinas (2006, ISSN:2389-8976) in sections 2.1 and 2.2. The saturated model is characterized in section 2.3 of that paper through the assumptions 1 and 2. Finally, the logistic model and its assumptions are explained in section 2.4.

Additionally, based on asymptotic theory for these ML-estimations and the score vector, the function lsm() calculates the values of the approximations for different deviations -2 log L, where L is the likelihood function. Based on these approximations, the function obtains the values of statistics for several hypothesis tests (each with an asymptotic chi-squared distribution): Null vs Logit, Logit vs Complete and Logit vs Saturated.

With the function lsm(), it is possible calculate confidence intervals for the logistic parameters and for the corresponding odds ratio. The asymptotic theory was developed for the case of independent, non-identically distributed variables. If Y is dichotomous and the data are grouped in J populations, it is recommended to use the function lsm() because it works very well for all K, the number of explanatory variables.

Usage

lsm(formula, family=binomial, data, na.action )

Arguments

formula

An expression of the form y ~ model, where y is the outcome variable (binary or dichotomous: its values are 0 or 1).

family

an optional family for defaul binomial

data

an optional data frame, list or environment (or object coercible by as.data.frame to a data frame) containing the variables in the model. If not found in data, the variables are taken from environment(formula), typically the environment from which lsm() is called.

na.action

a function which indicates what should happen when the data contain NAs.

Value

lsm returns an object of class "lsm".

An object of class "lsm" is a list containing at least the following components:

coefficients

Vector of coefficients estimations.

Std.Error

Vector of the coefficients<U+2019>s standard error.

Exp(B)

Vector with the exponential of the coefficients.

Wald

Value of the Wald statistic.

D.f

Degree of freedom for the Chi-squared distribution.

P.value

P-value with the Chi-squared distribution.

Log_Lik_Complete

Estimation of the log likelihood in the complete model.

Log_Lik_Null

Estimation of the log likelihood in the null model.

Log_Lik_Logit

Estimation of the log likelihood in the logistic model.

Log_Lik_Saturate

Estimation of the log likelihood in the saturate model.

Populations

Number of populations in the saturated model.

Dev_Null_vs_Logit

Value of the test statistic (Hypothesis: null vs logistic models).

Dev_Logit_vs_Complete

Value of the test statistic (Hypothesis: logistic vs complete models).

Dev_Logit_vs_Saturate

Value of the test statistic (Hypothesis: logistic vs saturated models).

Df_Null_vs_Logit

Degree of freedom for the test statistic<U+2019>s distribution (Hypothesis: null vs logistic models).

Df_Logit_vs_Complete

Degree of freedom for the test statistic<U+2019>s distribution (Hypothesis: logistic vs saturated models).

Df_Logit_vs_Saturate

Degree of freedom for the test statistic<U+2019>s distribution (Hypothesis: Logistic vs saturated models).

P.v_Null_vs_Logit

p-values for the hypothesis test: null vs logistic models.

P.v_Logit_vs_Complete

p-values for the hypothesis test: logistic vs complete models.

P.v_Logit_vs_Saturate

p-values for the hypothesis test: logistic vs saturated models.

Logit

Estimation of the logit function (the log-odds).

p_hat

Estimation of the probability that the outcome variable takes the value 1, given one population.

fitted.values

Vector with the values of the log_Likelihood in each jth population.

z_j

Vector with the values of each Zj (the sum of the observations in the jth population).

n_j

Vector with the nj (the number of the observations in each jth population).

p_j

Vector with the estimation of each pj (the probability of success in the jth population).

v_j

Vector with the variance of the Bernoulli variables in the jth population.

m_j

Vector with the expected values of Zj in the jth population.

V_j

Vector with the variances of Zj in the jth population.

V

Variance and covariance matrix of Z, the vector that contains all the Zj.

S_p

Score vector in the saturated model.

I_p

Information matrix in the saturated model.

Zast_j

Vector with the values of the standardized variable of Zj.

Details

The saturated model is characterized by the assumptions 1 and 2 presented in section 2.3 by Llinas (2006, ISSN:2389-8976).

References

[1] Humberto Jesus Llinas. (2006). Accuracies in the theory of the logistic models. Revista Colombiana De Estadistica,29(2), 242-244.

[2] Hosmer, D. (2013). Wiley Series in Probability and Statistics Ser. : Applied Logistic Regression (3). New York: John Wiley & Sons, Incorporated.

[3] Chambers, J. M. and Hastie, T. J. (1992) Statistical Models in S. Wadsworth & Brooks/Cole.

Examples

# NOT RUN {
# Hosmer, D. (2013) page 3: Age and coranary Heart Disease (CHD) Status of 20 subjects:

 AGE <- c(20, 23, 24, 25, 25, 26, 26, 28, 28, 29, 30, 30, 30, 30, 30, 30, 30, 32, 33, 33)
 CHD <- c(0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0)

 data <- data.frame (CHD, AGE)
 lsm(CHD ~ AGE , family = binomial,  data)

 # Other case.

    y <- c(1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1)
   x1 <- c(2, 2, 2, 5, 5, 5, 5, 8, 8, 11, 11, 11)
 
  data <- data.frame (y, x1)
  ELAINYS <-lsm(y ~ x1, family=binomial, data)
  summary(ELAINYS)

 ## For more ease, use the following notation.
  lsm(y~., family = binomial, data)

 ## Other case.

 y <- as.factor(c(1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1))
  x1 <- as.factor(c(2, 2, 2, 5, 5, 5, 5, 8, 8, 11, 11, 11))
 
  data <- data.frame (y, x1)
  ELAINYS1 <-lsm(y ~ x1, family=binomial, data)
  confint(ELAINYS1)
  
 ## For more ease, use the following notation.
  lsm(y~. , family = binomial, data)

# }