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, tm [0, 1], m 2, we de ne by induction the stochastic processes Xt1 ,..,tm = (Xt1 ,..,tm 1 )tm , Y t1 ,..,tm = (Y t1 ,..,tm 1 )tm , and the function t1 ,..,tm = ( t1 ,..,tm 1 )tm , using Lemma 76 for the computations at each stage With the aim of somewhat reducing the size of the formulas, we express the successive derivatives in terms of the processes Y t1 ,..,tm instead of Xt1 ,..,tm The reader must keep in mind that for each m-tuple t1 , , tm , the results depend on u through the expectation of the stochastic process Y t1 ,..,tm Also, for a stochastic process Z, we use the notation A(Z) = A0 (Z, ) = {Z(t) 0 : for all t [0, 1]} First Derivative.

Decisions must be made as to whether and how new entrants to the study population (births) should be included, whether a rotating design should be used, whether a repeated design should be used and, if so, whether the repeats should overlap or whether a split design is desirable These fundamental design options are discussed in 2 Additionally, some other features of sample design take on a rather different complex in the context of longitudinal surveys If face-to-face interviewing is to be used at any or.

Suppose that X satis es H2 We use formula (716) in Lemma 77 for 1, Z = X, and ( ) 1, obtaining for the rst derivative: F (u) = E[ 1A(Y 0 ) ]pY (0) (0) + E 1A(Y 1 ) pY (1) (0) I I.

(8.87)

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When the cat has learned to get out of the puzzle box by pulling a string, this does not mean that he has worked out how the string opens the door He would have learned just as well if the action had been quite alien to the reward, just as in the kind of learning studied by Pavlov Any arbitrary action that occurs just before a reward is more likely to be repeated A generation after Thorndike, Burrhus F Skinner5 developed his eponymous box, which is essentially a re ned and mechanized version of Thorndike s puzzle box The animal in the box presses a lever (if he is rat) or pecks a key (if he is a pigeon), and rewards or punishments are automatically delivered The times of all these events are continuously recorded.

The key idea of the model is that the relationship between Yp and Yp is modelled flexibly by a spline It is important to get that relationship right, since misspecifica tion of this aspect leads to bias Conditional on Yp , the distribution of the predictors is the same for respondents and nonrespondents, by the basic balancing property of the propensity score (Rosenbaum and Rubin, 1983) This means that misspecification of the form of g does not lead to bias for the mean of Yp given Yp (although it may lead to bias for other parameters) Thus for imputation, the form of g is less important than the form of the regression on Yp , although it is useful to include in g predictors of Yp that improve the precision of the imputations.

More generally, g should include variables with relationships with Yp that are of interest in the final analysis of the imputed data ^ ^ ^ l ML or REML estimates ; ; 2 ; ^ of the parameters ( , , 2,l) can be obtained using widely available software such as PROC MIXED in SAS (SAS, 2004) and lme( ) in S-plus (Pinheiro and Bates, 2000) Missing values of Yp can then be drawn from ^ ^ ^ l their predictive distribution of Yp given Yobs and ; ; 2 ; ^ This MI approach is improper (Rubin, 1987) in that it does not propagate uncertainty in estimating the parameters A proper MI procedure is to compute ML estimates of each set of imputations from a bootstrapped sample of cases.

Using his box, Skinner demonstrated the arbitrary nature of response learning in a most elegant experiment on superstition in the pigeon A hungry pigeon was put into the Skinner box and food was presented at regular intervals with no reference whatsoever to the bird s behavior After a short time the pigeon was seen repeatedly performing some arbitrary action One pigeon turned counter-clockwise around the box, making two or three turns between the appearances of food Another repeatedly thrust its head into one of the upper corners of the box A third developed a tossing response, as if placing its head beneath an invisible bar and lifting it repeatedly The pigeons had learned to repeat whatever action they happened to be performing just before the appearance of food.

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