^{1}

^{2}

^{3}

^{4}

^{1}

^{1}

^{2}

^{3}

^{4}

The complementary log-log is an alternative to logistic model. In many areas of research, the outcome data are continuous. We aim to provide a procedure that allows the researcher to estimate the coefficients of the complementary log-log model without dichotomizing and without loss of information. We show that the sample size required for a specific power of the proposed approach is substantially smaller than the dichotomizing method. We find that estimators derived from proposed method are consistently more efficient than dichotomizing method. To illustrate the use of proposed method, we employ the data arising from the NHSI.

Recently, logistic regression has become a popular tool in biomedical studies. The parameter in logistic regression has the interpretation of log odds ratio, which is easy for people such as physicians to understand. Probit and complementary log-log are alternatives to logistic model. For a covariate

These models use a categorical (dichotomous or polytomous) outcome variable. In many areas of research, the outcome data are continuous. Many researchers have no hesitation in dichotomizing a continuous variable, but this practice does not make use of within-category information. Several investigators have noted the disadvantages of dichotomizing both independent and outcome variables [

Moser and Coombs [

We aim to (a) provide a method that allows the researcher to estimate the coefficients of the complementary log-log model without dichotomizing and without loss of information, (b) show that the coefficient of the complementary log-log model can be interpreted in terms of the regression coefficients, (c) demonstrate that the coefficient estimates from this method have smaller variances and shorter confidence intervals than the dichotomizing method.

Let

the independent

Writing the model for each of the

the

The PDF and CDF of the extreme value distribution are given by

It is easy to check that

Let

We have assumed that

Let

We now compare the

The information matrix of generalized linear models has the form

Maximum likelihood estimation for the complementary log-log model is a special case of the generalized linear models. Let

then

In large samples,

By applying the delta method, let

In large samples, from (

By applying the delta method, let

In large samples,

In large samples,

where

the number of parameters

Total samples are

Table

Relative sample sizes required to attain any power for the dichotomizing method versus the proposed method.

Average proportion of successes ( | |||||

0.1 | 0.2 | 0.3 | 0.4 | 0.5 | |

0.25 | 23.7166 | 9.5092 | 7.4954 | 7.1996 | 6.8575 |

0.50 | 10.6719 | 5.4176 | 3.4215 | 2.5209 | 2.1784 |

0.75 | 7.7088 | 3.8713 | 2.5171 | 1.9380 | 1.5841 |

For given fixed

compute the value

calculate the cut-off point

As can be seen from Table

Here, we examine the relative efficiency of the estimate

Using (

It should be noted that these results hold true under the following assumptions:

the responses

the independent variables

For values of

It should be noted that, as in Table

Generate

For fixed

We simulated the data for two scenarios based on the distribution of the explanatory variable. In the first scenario, the independent variable follows a continuous uniform distribution and range (−2, 2), and in the second, the independent variable follows a truncated normal distribution with mean 0 and range (−2, 2). The relative mean square errors, relative interval lengths, absolute biases, and the probability of coverage were calculated.

Results of the simulations addressing the validity of the proposed method are displayed in Tables

Simulated relative mean square errors, relative intervals lengths, coverage probabilities, and absolute biases for the proposed and dichotomizing methods (using a continuous uniform distribution for the explanatory variable and an extreme value distribution for the errors).

Sample size | .75 | .9 | 1.1 | 1.2 | 1.3 | 1.4 | 1.5 | |
---|---|---|---|---|---|---|---|---|

1.15^{a} | 1.07 | 1.09 | 1.14 | 1.24 | 1.47 | 1.71 | ||

1.10^{b} | 1.03 | 1.03 | 1.07 | 1.14 | 1.23 | 1.35 | ||

0.10 | 0.943^{c} | 0.948 | 0.949 | 0.949 | 0.945 | 0.938 | 0.933 | |

0.948^{d} | 0.947 | 0.949 | 0.947 | 0.951 | 0.947 | 0.953 | ||

0.05^{e} | 0.04 | 0.12 | 0.14 | 0.10 | 0.15 | 0.11 | ||

0.07^{f} | 0.01 | 0.17 | 0.13 | 0.24 | 0.34 | 0.58 | ||

1.23 | 1.26 | 1.27 | 1.28 | 1.27 | 1.24 | 1.26 | ||

2.16 | 1.13 | 1.23 | 1.14 | 1.15 | 1.17 | 1.19 | ||

1000 | 0.50 | 0.940 | 0.951 | 0.951 | 0.945 | 0.942 | 0.937 | 0.934 |

0.951 | 0.949 | 0.951 | 0.950 | 0.948 | 0.947 | 0.948 | ||

0.04 | 0.01 | 0.08 | 0.10 | 0.05 | 0.09 | 0.04 | ||

0.05 | 0.04 | 0.15 | 0.12 | 0.09 | 0.12 | 0.13 | ||

12.75 | 12.44 | 13.22 | 12.68 | 13.14 | 12.91 | 12.79 | ||

3.67 | 3.57 | 3.58 | 3.63 | 3.69 | 3.76 | 3.84 | ||

0.95 | 0.943 | 0.951 | 0.952 | 0.944 | 0.944 | 0.938 | 0.929 | |

0.952 | 0.954 | 0.952 | 0.952 | 0.951 | 0.951 | 0.951 | ||

0.04 | 0.07 | 0.11 | 0.10 | 0.10 | 0.17 | 0.10 | ||

0.75 | 0.68 | 0.86 | 1.01 | 1.21 | 1.45 | 1.24 | ||

1.30 | 1.08 | 1.07 | 1.17 | 1.24 | 1.54 | 1.95 | ||

1.16 | 1.03 | 1.04 | 1.08 | 1.15 | 1.25 | 1.39 | ||

0.10 | 0.942 | 0.950 | 0.951 | 0.95 | 0.944 | 0.941 | 0.936 | |

0.951 | 0.950 | 0.949 | 0.951 | 0.954 | 0.954 | 0.953 | ||

0.12 | 0.07 | 0.24 | 0.25 | 0.21 | 0.18 | 0.29 | ||

0.23 | 0.08 | 0.33 | 0.39 | 0.41 | 0.73 | 1.21 | ||

1.35 | 1.10 | 1.27 | 1.26 | 1.26 | 1.25 | 1.26 | ||

1.26 | 1.03 | 1.13 | 1.14 | 1.16 | 1.17 | 1.20 | ||

500 | 0.50 | 0.940 | 0.949 | 0.947 | 0.948 | 0.943 | 0.940 | 0.933 |

0.952 | 0.951 | 0.949 | 0.949 | 0.954 | 0.950 | 0.951 | ||

0.23 | 0.34 | 0.27 | 0.23 | 0.26 | 0.25 | 0.38 | ||

0.48 | 0.11 | 0.17 | 0.18 | 0.31 | 0.26 | 0.42 | ||

13.04 | 13.17 | 13.8 | 13.90 | 14.45 | 14.48 | 14.47 | ||

3.72 | 3.65 | 3.68 | 3.73 | 3.82 | 3.91 | 3.99 | ||

0.95 | 0.942 | 0.947 | 0.951 | 0.949 | 0.947 | 0.938 | 0.935 | |

0.953 | 0.952 | 0.954 | 0.955 | 0.955 | 0.953 | 0.954 | ||

0.05 | 0.11 | 0.08 | 0.08 | 0.24 | 0.32 | 0.27 | ||

0.94 | 1.38 | 1.78 | 1.92 | 2.52 | 3.00 | 2.90 | ||

13.41 | 14.46 | 1.12 | 1.28 | 1.52 | 1.96 | 2.33 | ||

3.78 | 3.73 | 1.04 | 1.09 | 1.18 | 1.30 | 1.45 | ||

0.10 | 0.942 | 0.949 | 0.949 | 0.945 | 0.942 | 0.942 | 0.933 | |

0.957 | 0.954 | 0.948 | 0.949 | 0.952 | 0.957 | 0.953 | ||

0.02 | 0.20 | 0.38 | 0.33 | 0.42 | 0.41 | 0.66 | ||

2.11 | 2.74 | 0.42 | 0.84 | 1.18 | 1.78 | 2.24 | ||

1.27 | 1.25 | 1.32 | 1.28 | 1.30 | 1.30 | 1.29 | ||

1.16 | 1.13 | 1.13 | 1.14 | 1.16 | 1.18 | 1.20 | ||

250 | 0.50 | 0.941 | 0.948 | 0.952 | 0.947 | 0.945 | 0.943 | 0.933 |

0.951 | 0.951 | 0.951 | 0.950 | 0.951 | 0.951 | 0.951 | ||

0.12 | 0.13 | 0.35 | 0.44 | 0.41 | 0.53 | 0.55 | ||

0.11 | 0.22 | 0.39 | 0.47 | 0.51 | 0.74 | 0.59 | ||

12.98 | 14.6 | 15.64 | 15.46 | 17.05 | 16.89 | 18.33 | ||

3.75 | 3.72 | 3.82 | 3.88 | 4.01 | 4.12 | 4.29 | ||

0.95 | 0.945 | 0.955 | 0.946 | 0.948 | 0.940 | 0.937 | 0.932 | |

0.959 | 0.955 | 0.955 | 0.959 | 0.958 | 0.957 | 0.952 | ||

0.02 | 0.16 | 0.39 | 0.22 | 0.46 | 0.47 | 0.51 | ||

1.22 | 2.75 | 3.97 | 3.98 | 4.99 | 5.19 | 6.19 |

a: Relative mean square errors, b: Relative intervals lengths, c: Coverage probability (proposed), d: Coverage probability (dichotomized), e: % bias (proposed), f: % bias (dichotomized).

Simulated relative mean square errors, relative intervals lengths, coverage probabilities, and absolute biases for the proposed and dichotomizing methods (using a truncated normal distribution for the explanatory variable and an extreme value distribution for the errors).

Sample size | .75 | .9 | 1.1 | 1.2 | 1.3 | 1.4 | 1.5 | |
---|---|---|---|---|---|---|---|---|

1.17^{a} | 1.02 | 1.08 | 1.13 | 1.19 | 1.28 | 1.36 | ||

1.11^{b} | 1.03 | 1.03 | 1.06 | 1.10 | 1.25 | 1.22 | ||

0.10 | 0.942^{c} | 0.948 | 0.948 | 0.952 | 0.944 | 0.942 | 0.940 | |

0.951^{d} | 0.951 | 0.950 | 0.952 | 0.949 | 0.951 | 0.951 | ||

0.08^{e} | 0.06 | 0.03 | 0.14 | 0.13 | 0.14 | 0.16 | ||

0.10^{f} | 0.11 | 0.15 | 0.23 | 0.30 | 0.39 | 0.39 | ||

1.26 | 1.24 | 1.26 | 1.28 | 1.28 | 1.25 | 1.28 | ||

1.24 | 1.13 | 1.13 | 1.14 | 1.14 | 1.15 | 1.17 | ||

1000 | 0.50 | 0.944 | 0.948 | 0.952 | 0.947 | 0.947 | 0.944 | 0.941 |

0.948 | 0.951 | 0.949 | 0.949 | 0.947 | 0.950 | 0.949 | ||

0.02 | 0.09 | 0.08 | 0.07 | 0.18 | 0.16 | 0.13 | ||

0.03 | 0.06 | 0.12 | 0.16 | 0.20 | 0.16 | 0.14 | ||

12.33 | 13.12 | 13.03 | 12.71 | 12.86 | 12.55 | 12.88 | ||

3.62 | 3.59 | 3.61 | 3.62 | 3.64 | 3.68 | 3.71 | ||

0.95 | 0.944 | 0.951 | 0.948 | 0.948 | 0.945 | 0.945 | 0.946 | |

0.952 | 0.948 | 0.95 | 0.949 | 0.949 | 0.951 | 0.952 | ||

0.10 | 0.04 | 0.11 | 0.04 | 0.16 | 0.16 | 0.20 | ||

1.26 | 1.05 | 1.56 | 1.36 | 1.43 | 1.80 | 1.94 | ||

1.18 | 1.09 | 1.06 | 1.75 | 1.23 | 1.32 | 1.58 | ||

1.11 | 1.03 | 1.03 | 1.06 | 1.11 | 1.16 | 1.23 | ||

0.10 | 0.945 | 0.95 | 0.951 | 0.951 | 0.949 | 0.943 | 0.944 | |

0.953 | 0.953 | 0.953 | 0.950 | 0.949 | 0.951 | 0.950 | ||

0.04 | 0.13 | 0.31 | 0.18 | 0.33 | 0.36 | 0.37 | ||

0.21 | 0.08 | 0.37 | 0.50 | 0.62 | 0.69 | 0.96 | ||

1.25 | 1.27 | 1.27 | 1.29 | 1.27 | 1.29 | 1.25 | ||

1.14 | 1.13 | 1.13 | 1.14 | 1.15 | 1.16 | 1.17 | ||

500 | 0.50 | 0.944 | 0.948 | 0.949 | 0.947 | 0.948 | 0.944 | 0.935 |

0.951 | 0.951 | 0.951 | 0.948 | 0.951 | 0.948 | 0.949 | ||

0.13 | 0.22 | 0.35 | 0.37 | 0.35 | 0.30 | 0.44 | ||

0.16 | 0.19 | 0.39 | 0.48 | 0.44 | 0.41 | 0.54 | ||

13.11 | 14.02 | 14.02 | 13.5 | 13.54 | 13.80 | 14.32 | ||

3.73 | 3.71 | 3.73 | 3.75 | 3.77 | 3.81 | 3.86 | ||

0.95 | 0.944 | 0.95 | 0.951 | 0.950 | 0.947 | 0.944 | 0.944 | |

0.954 | 0.95 | 0.951 | 0.953 | 0.948 | 0.956 | 0.953 | ||

0.15 | 0.10 | 0.24 | 0.38 | 0.32 | 0.33 | 0.43 | ||

2.50 | 2.70 | 2.92 | 3.10 | 2.92 | 3.36 | 3.89 | ||

1.28 | 1.11 | 1.12 | 1.19 | 1.33 | 1.54 | 1.76 | ||

1.11 | 1.03 | 1.04 | 1.08 | 1.13 | 1.19 | 1.28 | ||

0.10 | 0.947 | 0.951 | 0.950 | 0.947 | 0.950 | 0.950 | 0.942 | |

0.951 | 0.950 | 0.950 | 0.952 | 0.954 | 0.952 | 0.951 | ||

0.40 | 0.34 | 0.37 | 0.64 | 0.69 | 0.58 | 0.81 | ||

0.26 | 0.06 | 0.69 | 1.08 | 1.30 | 1.55 | 2.22 | ||

1.32 | 1.30 | 1.27 | 1.33 | 1.31 | 1.33 | 1.31 | ||

1.15 | 1.13 | 1.13 | 1.14 | 1.18 | 1.17 | 1.18 | ||

250 | 0.50 | 0.951 | 0.95 | 0.953 | 0.951 | 0.940 | 0.945 | 0.940 |

0.949 | 0.951 | 0.952 | 0.948 | 0.948 | 0.950 | 0.948 | ||

0.22 | 0.43 | 0.57 | 0.69 | 0.66 | 0.58 | 0.66 | ||

0.38 | 0.53 | 0.64 | 0.89 | 0.91 | 0.82 | 0.91 | ||

14.09 | 14.51 | 16.27 | 15.91 | 15.89 | 15.73 | 15.60 | ||

3.86 | 3.87 | 3.93 | 3.92 | 3.98 | 4.04 | 4.11 | ||

0.95 | 0.943 | 0.95 | 0.951 | 0.951 | 0.947 | 0.944 | 0.937 | |

0.953 | 0.95 | 0.953 | 0.956 | 0.953 | 0.956 | 0.952 | ||

0.30 | 0.37 | 0.57 | 0.68 | 0.42 | 0.62 | 0.75 | ||

4.98 | 5.52 | 6.547 | 5.91 | 6.17 | 6.88 | 7.72 |

a: Relative mean square errors, b: Relative intervals lengths, c: Coverage probability (proposed), d: Coverage probability (dichotomized), e: % bias (proposed), f: % bias (dichotomized).

The simulations show that the relative mean square errors are all greater than 1, increasing with the average proportion of successes and when the

To illustrate the application of the proposed method presented in the previous section, we utilize the data arising from the National Health Survey in Iran. The other analyses using this data appear in many places [

In this study, 14176 women aged 20–69 years were investigated. BMI (body mass index), our dependent variable, was calculated as weight in kilograms divided by height in meters squared (kg/m^{2}). Independent variables included place of residence, age, smoking, economic index, marital status, and education level. The independent variables considered were both categorical and continuous. At first, BMI was treated as a continuous variable, and ^{2}) and nonobese (BMI <30 kg/m^{2}). A complementary log-log model was used for the binary analysis, with obese or nonobese used as the outcome measure. The

Adjusted

Covariates | 95% CI^{a} (proposed) | 95% CI (dichotomized) | Relative^{b} length of CI | |
---|---|---|---|---|

Place of residence | 1.65 (1.97)^{c} | 1.58–1.74 | 1.79–2.18 | 2.43 |

Age | 1.021 (1.019) | 1.018–1.022 | 1.015–1.022 | 1.75 |

Years of education | 0.99 (0.98) | 0.985–0.997 | 0.971–0.994 | 1.92 |

Smoking | 0.76 (0.68) | 0.66–0.90 | 0.51–0.92 | 1.71 |

Marital status | 1.16 (1.42) | 1.10–1.22 | 1.27–1.58 | 2.58 |

Lower-middle economy index | 1.24 (1.32) | 1.14–1.32 | 1.18–1.48 | 1.67 |

Upper-middle economy index | 1.21 (1.26) | 1.14–1.29 | 1.12–1.42 | 2.0 |

High economy index | 1.20 (1.21) | 1.11–1.30 | 1.08–1.36 | 1.47 |

^{
a}confidence interval, ^{b}dichotomized/proposed, ^{c}proposed (dichotomized).

When assuming the errors

Our results were consistent with the findings by Moser and Coombs [

Our main recommendation is to let continuous response remain continuous. Do not throw away information by transforming the data to binary. This means that if the objective is to estimate and/or test coefficients when responses are continuous, please resist dichotomizing your response variable.

(a)

It is easy to check that

(b)

(c)

The authors have declared no conflict of interests.