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In this chapter and the looking for quick easy and discrete, we present statistical models of hazard for data collected in discrete time. In subsequent chapters, we extend these basic ideas to situations in which event occurrence is recorded in continuous time. Good data analysis involves more than using a computer package to fit a statistical model to data. To conduce a credible discrete-time survival analysis, you must: 1 specify a suitable model for hazard and understand its assumptions; 2 use sample data to estimate the model parameters; 3 interpret in terms of your research questions; 4 evaluate model fit and [express the uncertainty in the] model parameters; and 5 communicate your findings.
In this chapter and the next, we present statistical models of hazard for data collected in discrete time.
In subsequent chapters, we extend these basic ideas to situations in which event occurrence is recorded in continuous time. Good data analysis involves more than using a computer package to fit a statistical model to data. To conduce a credible discrete-time survival analysis, you must: 1 specify a suitable model for hazard and understand its assumptions; 2 use sample data to estimate the model parameters; 3 interpret in terms of your research questions; 4 evaluate model fit and [express the uncertainty in the] model looking for quick easy and discrete and 5 communicate your findings.
Since these data show no censoring before the final time point, it is straightforward to follow along with the text p.
Plots of sample hazard functions and survivor functions estimates separately for groups distinguished by their predictor values are invaluable exploratory tools. If discrehe predictor is categorical, like PTconstruction of these displays is straightforward. If a predictor is continuous, you should just temporarily categorize its values for plotting purposes. Looking for quick easy and discrete make our version of the descriptive plots in Figure With fit Before we plot the foe, it might be handy to arrange the fit in life tables.
This will be our version of Table Combine the two ggplot2 objects with patchwork syntax to make our version of Figure OnSinger and Willett compared the hazard probabilities at grades 8 and 11 for boys in the two pt groups.
We can make that comparison with filter. At the top ofSinger and Willett compared the percentages of boys who were virgins at grades 9 and 12, by pt status. Those percentages are straight algebraic transformations of the corresponding survival function values. To postulate a statistical model to represent the relationship between the population discrete-time hazard function and predictors, we must deal with two complications apparent in these displays.
One is that any hypothesized model must describe the shape of the entire discrete-time hazard function over time, not just its value in any one period, in much the same eazy that a multilevel model for change characterizes the shape of entire individual growth trajectories over lookign. A second complication is that, as a conditional probability, the value of discrete-time hazard must lie between 0 and looking for quick easy and discrete. Any reasonable statistical model for hazard must recognize this constraint, precluding the occurrence of theoretically impossible values.
A conventional way to handle the bounded nature of probabilities is transform the scale of the data. Cox recommended either the odds and log-odds i. Log-odds is a minor extension; you simply take the log of the odds, which we can formally express as. Before we make our version of Figure Odds are bounded to values of zero and above and have an inflection at 1. Log-odds are unbounded and have an inflection point at 0. With the survival models from the prior sections, we were lazy and just used the survival package.
But recall from the end of the last chapter that we can fit the analogous models brms using the binomial likelihood. This subsection is a great place llooking practice those some more. The fitted lines Singer and Willett displayed in Figure However, the sex data looking for quick easy and discrete not in a convenient form to fit those models. Sure, their focus was on the frequentist approach using maximum likelihood.
But the point still stands. If these model fitting lokoing feel a bit rushed, they are. Any anxious feelings aside, now fit the three binomial models.
We continue to use weakly-regularizing priors for each. We can extract the fitted values and their summaries for each row in the data with fitted. In addition to the posterior means i. Singer and Willet mused the unconstrained model fit6 was a better fit to the data than the other two. We can quantify that with a LOO comparison. Earlier equations for the hazard function omitted substantive predictors.
We can use this to define the conditional hazard function as. We can express the discrete conditional hazard model with a general functional form with respect to time as. This is just the type of model we used to fit fit For that model, the basic equation was.
However, that formula could be a little misleading. Recall the formula:. Both were saved as factor variables. The same basic thing goes for pt. Because pt was a factor used in a model formula with no conventional intercept, it acted as if it was a series of 2 dummy variables with no reference category. Thus, we might rewrite the model equation for fit6 as.
And what is the criterion, event? Given our factor coding of ptour two submodels for the equations in the last section are. Discrege we consider the generic discrete conditional hazard function, that would follow the form. This is just a particular kind of logistic regression model. We can explore what that might look like with our version of Figure Those were:.
At the beginning of section It has our substantive predictors pt and pastoo. Bayesian logistic regression via the binomial likelihood has been dor approach. In one sense, fitting discrete-hazard models with Bayesian logistic regression looking for quick easy and discrete old hat, for us. To show what I mean, we might look at the data we used for our last model, fit Instead, each event cell only takes on a value of 0 or 1 i.
Whether you are working with aggregated or un-aggregated data, both are suited to fit logistic regression models with the binomial likelihood. Just specify the necessary information in the model syntax. For brmsthe primary difference is how you use the trials function. In the case of unaggregated binomial data, we can just state trials 1. When viewed in bulk, all those print calls yield a lot of output.
We can arrange the parameter summaries similar to those in Table The amount and direction anf variation in their values describe the shape of this function and tell us whether risk increases, decreases, or remains steady over time. Because they were in the log-odds scale, the model output and our coefficient plot can be difficult to interpret. If we take the anti-log i.
This tells us that, in every grade, the estimated odds of first intercourse are nearly two and one half times higher for boys who looking for quick easy and discrete lookinb parenting transition in comparison to boys raised with both biological parents. In substantive terms, an odds ratio of this magnitude represents a substantial, and potentially important, effect.
Disfrete reframe the odds ratio in terms of the other group i. To understand pasour measure of parental antisocial behavior, it will help to look at its range. Exponentiating i. However, this creates a minor challenge. The catch is, we need to make sure that random value is constant for each case.
Our solution will llooking to first nest the data such that each case only has one row. We can use the tidy function and a few lines of wrangling code to make a version of the table looking for quick easy and discrete the middle of Because our data did not include the original values for pt1 through pt3the in our table will not match those in the text.
We did get pretty close, though, eh? Hopefully this gives a sense of the workflow. We can make our version of Table To reduce clutter, we will use abbreviated column names.
For the alpha and beta columns, we just subset the values looking for quick easy and discrete fixef. The two logit-hazard columns, lh0 and lh1were simple algebraic transformations of alpha and betarespectively. To make the two survival columns, s0 and s1we applied the cumprod function to one minus the two hazard columns. Note how all this is based off of the posterior means. Two of these plots are quite similar to two of the subplots from Figure But recall that though those plots were based on fit Regardless of whether the logistic regression model is based on aggregated data, the post-processing approach will involve the fitted function.
However, the specifics of how we use fitted will differ. For the disaggregated data used to fit fit The values for grade 6 i. Here is the breakdown of what percentage of boys will still be virgins at grades 9 and 12, based on pt status, as indicated by fit It is easy to display fitted hazard and survivor functions for model involving multiple predictor by extending these ideas in a straightforward manner.
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