A few remarks on a small example by Jean-Claude Deville regarding non-ignorable non-response Section 2. NotationA few remarks on a small example by Jean-Claude Deville regarding non-ignorable non-response Section 2. Notation

Table 2.1 shows the notation for Table 1.1.

Table 2.1
Notation for Table 1.1
Table summary
This table displays the results of Notation for Table 1.1. The information is grouped by (appearing as row headers), Drug User, Non-user, Missing and Total (appearing as column headers).
	Drug User	Non-user	Missing	Total
Male	$r_{H D}$	$r_{H S}$	$m_{H}$	$n_{H .}$
Female	$r_{F D}$	$r_{F S}$	$m_{F}$	$n_{F .}$
Total	$r_{. D}$	$r_{. S}$	$m$	$n$

For simplicity, assume that we are dealing with a census. In other words, the 600 students were not randomly selected. Therefore, the only source of randomness is the non-response mechanism. This assumption is not that restrictive, since it is equivalent to considering that the sample is random, but that the reasoning below is conditional on the random sample. The objective is to estimate the numbers of people in Table 2.2. This table is assumed not to be random. It is therefore a matter of distributing the non-respondents $m_{H}$ and $m_{F}$ between drug users and non-users.

Table 2.2
Number of people to be estimated based on Table 1.1
Table summary
This table displays the results of Number of people to be estimated based on Table 1.1. The information is grouped by (appearing as row headers), Drug User, Non-user and Total (appearing as column headers).
	Drug User	Non-user	Total
Male	$n_{H D}$	$n_{H S}$	$n_{H .}$
Female	$n_{F D}$	$n_{F S}$	$n_{F .}$
Total	$n_{. D}$	$n_{. S}$	$n$

As well, it is assumed that the non-response follows a Poisson design, that is, each individual decides whether or not to respond with a probability independent of other individuals. The response probability may vary among individuals.

The two vectors $(r_{H D}, r_{H S}, m_{H}),$ and $(r_{F D}, r_{F S}, m_{F})$ each have a multinomial distribution whose parameters depend on the model used. MCAR cases, which are completely trivial, will not be studied. In Table 2.3, which shows cases of MAR, the response probability depends on gender only $(p_{H}$ for males, $p_{F}$ for females). In Table 2.4, which shows cases of NMAR, the response probability depends only on being or not being a drug user $(q_{D},$ $q_{S}$ for the others).

Table 2.3
Case 1: MAR model, non-response depends on gender
Table summary
This table displays the results of Case 1: MAR model. The information is grouped by (appearing as row headers), Drug User , Non-user , Missing and Total (appearing as column headers).
	Drug User	Non-user	Missing	Total
Male	$E (r_{H D}) = n_{H D} p_{H}$	$E (r_{H S}) = n_{H S} p_{H}$	$E (m_{H}) = n_{H .} (1 - p_{H})$	$n_{H .}$
Female	$E (r_{F D}) = n_{F D} p_{F}$	$E (r_{F S}) = n_{F S} p_{F}$	$E (m_{F}) = n_{F .} (1 - p_{F})$	$n_{F .}$
Total	$E (r_{. D})$	$E (r_{. S})$	$m$	$n$

Table 2.4
Case 2: NMAR model, non-response depends on being or not being a drug user
Table summary
This table displays the results of Case 2: NMAR model. The information is grouped by (appearing as row headers), Drug User , Non-user, Missing and Total (appearing as column headers).
	Drug User	Non-user	Missing	Total
Male	$E (r_{H D}) = n_{H D} q_{D}$	$E (r_{H S}) = n_{H S} q_{S}$	$E (m_{H}) = n_{H D} (1 - q_{D}) + n_{H S} (1 - q_{S})$	$n_{H .}$
Female	$E (r_{F D}) = n_{F D} q_{D}$	$E (r_{F S}) = n_{F S} q_{S}$	$E (m_{F}) = n_{F D} (1 - q_{D}) + n_{F S} (1 - q_{S})$	$n_{F .}$
Total	$E (r_{. D})$	$E (r_{. S})$	$m$	$n$

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A few remarks on a small example by Jean-Claude Deville regarding non-ignorable non-response Section 2. NotationA few remarks on a small example by Jean-Claude Deville regarding non-ignorable non-response Section 2. Notation