**
****
**Dogs: loglinear model for

binary data

Lindley (19??) analyses data from Kalbfleisch (1985) on the Solomon-Wynne experiment on dogs, whereby they learn to avoid an electric shock. A dog is put in a compartment, the lights are turned out and a barrier is raised, and 10 seconds later an electric shock is applied. The results are recorded as success (Y = 1 ) if the dog jumps the barrier before the shock occurs, or failure (Y = 0) otherwise.

Thirty dogs were each subjected to 25 such trials. A plausible model is to suppose that a dog learns from previous trials, with the probability of success depending on the number of previous shocks and the number of previous avoidances. Lindley thus uses the following model

p
_{j
} = A
^{x
}^{j
} B
^{j-x
}^{j
}

for the probability of a shock (failure) at trial j, where x
_{j
}_{
}= number of success (avoidances) before trial j and j - x
_{j
}_{
}= number of previous failures (shocks). This is equivalent to the following log linear model

log
p
_{j
} =
a
x
_{j
}
+ b
( j-x
_{j
}_{
})

Hence we have a generalised linear model for binary data, but with a log-link function rather than the canonical logit link. This is trivial to implement in BUGS:

model

{

for (i in 1 : Dogs) {

xa[i, 1] <- 0; xs[i, 1] <- 0 p[i, 1] <- 0

for (j in 2 : Trials) {

xa[i, j] <- sum(Y[i, 1 : j - 1])

xs[i, j] <- j - 1 - xa[i, j]

log(p[i, j]) <- alpha * xa[i, j] + beta * xs[i, j]

y[i, j] <- 1 - Y[i, j]

y[i, j] ~ dbern(p[i, j])

}

}

alpha ~ dflat()T(, -0.00001)

beta ~ dflat()T(, -0.00001)

A <- exp(alpha)

B <- exp(beta)

}

__Data
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( click to open )

__Inits for chain 1____
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__Inits for chain 2____
__ ( click to open )

Results