Extending the model to account for missing data

Correcting for Survey Misreports using Auxiliary Information with an Application to Estimating Turnout 40

4.2 Correcting for misreporting in binary choice models .1 Defining the problem

4.2.3 Extending the model to account for missing data

parametric versus non-parametric methods to correct for covariate-dependent misclassification and evaluating the relative weaknesses and advantages of both approaches in applied work.

other hand, Bayesian methods such as the one presented in this paper can easily accommodate missing data. There is no distinction between missing data and parameters within the Bayesian framework, and thus inference in this setting essentially requires defining a prior for the missing values and sampling from the joint posterior distribution of the parameters and missing values, incorporating just an “extra-layer” in the Gibbs sampling algorithm compared to the complete-case analysis (Gelman et al. 2004; Ibrahim et al. 2005). In particular, our model can be immediately extended to deal with missing response and covariate values, including cases with missing responses alone, with missing covariates alone, and with missing covariates and responses. This allows us to accommodate item and unit nonresponse in both the main and the validation studies.⁵⁸

Let ^{w =}ⁱ

(

^wⁱ^,1^,^wⁱ^,2^,...^w^{i p}^,

)

^'^{, 1,..,}ⁱ⁼ ^N denote a p×1 vector of covariates included in xi, z¹_i and z_i², and denote the marginal density of w_i by p

(

wi^|α

)

, where α parametrizes the joint distribution of the covariates. Adopting the notation in Chen et al.

(2008), we write w = w_i

(

_{i obs}, ,w_{i mis},

)

, where w_{i mis}_, is the q_i×1 vector of missing components of w_i, 0≤ ≤q_i p, and w_{i obs}_, is the observed portion of w_i. Similarly, we use

_{i mis}_,

y if the self-reported outcome 

yi is missing, and y_{i obs}_, otherwise. We assume that the missing data mechanism is ignorable (Rubin 1976; Little and Rubin 2002). That is, we assume that the missing data mechanism does not depend on the missing values, but may depend on the observed outcome and covariate data included in the model - i.e., the data

58 However, as seen in Equation 3.11 below, respondents with completely missing outcomes and covariates do not contribute to the likelihood function.

are missing at random (MAR) - and that the parameters governing the missing data mechanism are distinct from the parameters of the sampling model. The observed-data likelihood for the main study can then be written as:

( ) (

^

)



( )

(



)

 ( )

( )

(



)



, ,

1 2 1 2

1 2 , , ,

, ,

1 2

, , | | , , , |

| , , , , | | , , ,

i i obs i obs

i i obs i mis i obs

i mi

obs i i i

i i obs i mis i mis

i y

i i mis y

L p y p

p y p d

p y

γ γ β γ γ β α

γ γ β α

γ γ β

= ×

∏

∏ ∫

∫



w w

w w w

w w

w w w w

( )

^

(



)

 ( )

( )

^

(



)



, ,

, 1 2 , , ,

, ,

1 2

, ,

| , , , , | | , , ,

i i obs s

i i obs i mis i mis

i i m

i mis

i i mis

i mis i obs i mis i mis

i mis i mis

i i mis y

p d y

p y p d y d

p y

γ γ β α

γ γ β

∏

∏ ∫∫

∫∫

w w

w w w

w w

w w w w

(

^, ^, ^, ^|

)

^^, ^,

i obs i mis i mis i mis

p α d y d

∏

^w ^w ^w

(4.10)

which, as noted by Chen et al. (2008), reduces to:

( ) (

^

)



( )

(



)

 ( )

( )



, ,

1 2 1 2

1 2 , , ,

, ,

, , | | , , , |

| , , , , |

i i obs

i obs

i i obs i mis

i obs

i i obs

i mis

obs i i i

i i obs i mis i mis

i y

L p y p

p y p d

γ γ β γ γ β α

γ γ β α

= ×

∏

∏ ∫

∏



w w

w w w

w w

w w w w

( )

, ( , , )

, , ,

, ,

, |

i i obs i mis

i mis

i obs i mis i mis

p α d

∏

w w w

(4.11).

As suggested by Ibrahim, Chen and Lipsitz (2002), it is often convenient to model the joint distribution p

(

^wi^|α

)

as a series of one-dimensional conditional distributions:

( ) ( )

,1 ,2 , , ,1 ,2 , 1

, 1 ,1 ,2 , 2 1 ,1 1

, ,... | | , ,... ,

| , ,... , ... |

i i i p i p i i i p p

i p i i i p p i

p w w w p w w w w

p w w w w p w

α α

−

− − −

= ×

× ×

(4.12),

where α_l, 1,.., ,l= p is a vector of parameters for the llth conditional distribution, the α_l 's are distinct, and α =

(

α α1, 2,...,α_p

)

. As noted by these authors, specification 4.12 has the advantages of easing the prior elicitation for α and reducing the computational burden of the Gibbs algorithm required for sampling from the observed data posterior, and is particularly well-suited for cases in which w includes categorical and continuous covariates. While the modeling of the covariate distributions depends on the order of the conditioning, Ibrahim, Chen and Lipsitz (2002) show that posterior inferences are generally quite robust to changes in the order of the conditioning. Obviously, 4.12 needs to be specified only for those covariates that have missing values. If some of the covariates in w are completely observed for all respondents in a survey, they can be conditioned on when constructing the distribution of the missing covariates.

The joint posterior density of the unknown parameters based on the observed data is then given by:

(

γ γ β α1, 2, , |_obs

)

∝L

(

γ γ β α1, 2, , |_obs

) (

×p γ γ β α1, 2, ,

)

(4.13).

Information on the misreport patterns and on all the parameters of interest can be incorporated from the validation study in essentially identical way as in the case with no missing data. A joint prior for

(

γ γ β α1, 2, ,

)

could be specified as:

(

γ γ β α1, 2, ,

)

∝L

(

γ γ β α1, 2, , |_obs

)

^δ×p

( ) ( ) ( ) ( )

γ1 ×p γ2 ×p β ×p α

(4.14),

where ^L

(

γ γ β α₁^, ₂^{, ,} ^|_obs

)

is obtained from the complete-data likelihood of the validation study:

(

^{γ γ β α}¹^, ²^{, ,} ^|^^obs

)

⁼

_∫∫

(

^^{y y w}^{, |} ^{, ,}^{γ γ β α}¹ ²^{, ,}

)

^{d y}^^mis^d^w^mis (4.15) and, as mentioned above, δ is a scalar prior parameter that weights the validated data relative to the data from the main study.⁵⁹

i

Note that our specification allows for missing

responses and covariate values in the validated sample as well, and can accommodate cases in which the missing self-reported variable depends on the true y_i. As in the case of no missing data, it is also possible to incorporate only the information from the observed probability of misreporting in the validation study to specify the priors for γ₁, γ₂ and a subset α_z of the components of α for the main study, while using diffuse prior distributions for the remaining parameters. However, the additional information obtained from L

(

γ γ β α1, 2, , |_obs

)

can increase efficiency in many missing data problems in which certain parameters in the likelihood function are not identifiable and/or very little information is available for inference, particularly when the “gold-standard” measure y_i is observed for a large proportion of the respondents in the validation study (Ibrahim, Chen and Lipsitz 2002; Reilly and Pepe 1995; Robins, Rotnitzky and Zhao 1994).

59 See Section 4 in Ibrahim, Chen and Lipsitz (2002) for details.

In principle, it is possible to extend this approach to the case of non-ignorably missing values following Huang, Chen and Ibrahim (1999), Ibrahim and Lipsitz (1996) and Ibrahim, Lipsitz and Chen (1999). However, there is usually little information on the missing data mechanism, and the parameters of the missing data model are often quite difficult to estimate (Ibrahim, Lipsitz and Horton 2001). The plausibility of the MAR assumption can be enhanced by including additional individual and contextual variables in the model specification (Gelman et al. 2004; Gelman, King and Liu 1998).

Dalam dokumen ESSAYS IN ELECTORAL BEHAVIOR AND BAYESIAN DATA ANALYSIS Thesis by Gabriel Katz In Partial Fulfillment of the Requirements for the degree of Doctor of Philosophy (Halaman 135-140)