# Power and Sample Size for Ordinal Response `popower` computes the power for a two-tailed two sample comparison of ordinal outcomes under the proportional odds ordinal logistic model. The power is the same as that of the Wilcoxon test but with ties handled properly. `posamsize` computes the total sample size needed to achieve a given power. Both functions compute the efficiency of the design compared with a design in which the response variable is continuous. `print` methods exist for both functions. Any of the input arguments may be vectors, in which case a vector of powers or sample sizes is returned. These functions use the methods of Whitehead (1993). `pomodm` is a function that assists in translating odds ratios to differences in mean or median on the original scale. `simPOcuts` simulates simple unadjusted two-group comparisons under a PO model to demonstrate the natural sampling variability that causes estimated odds ratios to vary over cutoffs of Y. `propsPO` uses `ggplot2` to plot a stacked bar chart of proportions stratified by a grouping variable (and optionally a stratification variable), with an optional additional graph showing what the proportions would be had proportional odds held and an odds ratio was applied to the proportions in a reference group. If the result is passed to `ggplotly`, customized tooltip hover text will appear. `propsTrans` uses `ggplot2` to plot all successive transition proportions. `formula` has the state variable on the left hand side, the first right-hand variable is time, and the second right-hand variable is a subject ID variable.\ `multEventChart` uses `ggplot2` to plot event charts showing state transitions, account for absorbing states/events. It is based on code written by Lucy D'Agostino McGowan posted at [https://livefreeordichotomize.com/posts/2020-05-21-survival-model-detective-1/](https://livefreeordichotomize.com/posts/2020-05-21-survival-model-detective-1/). ```r popower(p, odds.ratio, n, n1, n2, alpha=0.05) ## S3 method for class 'popower' print(x, ...) posamsize(p, odds.ratio, fraction=.5, alpha=0.05, power=0.8) ## S3 method for class 'posamsize' print(x, ...) pomodm(x=NULL, p, odds.ratio=1) simPOcuts(n, nsim=10, odds.ratio=1, p) propsPO(formula, odds.ratio=NULL, ref=NULL, data=NULL, ncol=NULL, nrow=NULL ) propsTrans(formula, data=NULL, labels=NULL, arrow='\u2794', maxsize=12, ncol=NULL, nrow=NULL) multEventChart(formula, data=NULL, absorb=NULL, sortbylast=FALSE, colorTitle=label(y), eventTitle='Event', palette='OrRd', eventSymbols=c(15, 5, 1:4, 6:10), timeInc=min(diff(unique(x))/2)) ``` ## Arguments - `p`: a vector of marginal cell probabilities which must add up to one. For `popower` and `posamsize`, The `i`th element specifies the probability that a patient will be in response level `i`, averaged over the two treatment groups. For `pomodm` and `simPOcuts`, `p` is the vector of cell probabilities to be translated under a given odds ratio. For `simPOcuts`, if `p` has names, those names are taken as the ordered distinct Y-values. Otherwise Y-values are taken as the integers 1, 2, ... up to the length of `p`. - `odds.ratio`: the odds ratio to be able to detect. It doesn't matter which group is in the numerator. For `propsPO`, `odds.ratio` is a function of the grouping (right hand side) variable value. The value of the function specifies the odds ratio to apply to the refernce group to get all other group's expected proportions were proportional odds to hold against the first group. Normally the function should return 1.0 when its `x` argument corresponds to the `ref` group. For `pomodm` and `simPOcuts` is the odds ratio to apply to convert the given cell probabilities. - `n`: total sample size for `popower`. You must specify either `n` or `n1` and `n2`. If you specify `n`, `n1` and `n2` are set to `n/2`. For `simPOcuts` is a single number equal to the combined sample sizes of two groups. - `n1`: for `popower`, the number of subjects in treatment group 1 - `n2`: for `popower`, the number of subjects in group 2 - `nsim`: number of simulated studies to create by `simPOcuts` - `alpha`: type I error - `x`: an object created by `popower` or `posamsize`, or a vector of data values given to `pomodm` that corresponds to the vector `p` of probabilities. If `x` is omitted for `pomodm`, the `odds.ratio` will be applied and the new vector of individual probabilities will be returned. Otherwise if `x` is given to `pomodm`, a 2-vector with the mean and median `x` after applying the odds ratio is returned. - `fraction`: for `posamsize`, the fraction of subjects that will be allocated to group 1 - `power`: for `posamsize`, the desired power (default is 0.8) - `formula`: an R formula expressure for `proposPO` where the outcome categorical variable is on the left hand side and the grouping variable is on the right. It is assumed that the left hand variable is either already a factor or will have its levels in the right order for an ordinal model when it is converted to factor. For `multEventChart` the left hand variable is a categorial status variable, the first right hand side variable represents time, and the second right side variable is a unique subject ID. One line is produced per subject. - `ref`: for `propsPO` specifies the reference group (value of the right hand side `formula` variable) to use in computing proportions on which too translate proportions in other groups, under the proportional odds assumption. - `data`: a data frame or `data.table` - `labels`: for `propsTrans` is an optional character vector corresponding to y=1,2,3,... that is used to construct `plotly` hovertext as a `label` attribute in the `ggplot2` aesthetic. Used with y is integer on axes but you want long labels in hovertext. - `arrow`: character to use as the arrow symbol for transitions in `propsTrans. The default is the dingbats heavy wide-headed rightwardsarror.` - `nrow,ncol`: see `facet_wrap` - `maxsize`: maximum symbol size - ``...``: unused - `absorb`: character vector specifying the subset of levels of the left hand side variable that are absorbing states such as death or hospital discharge - `sortbylast`: set to `TRUE` to sort the subjects by the severity of the status at the last time point - `colorTitle`: label for legend for status - `eventTitle`: label for legend for `absorb` - `palette`: a single character string specifying the `scale_fill_brewer` color palette - `eventSymbols`: vector of symbol codes. Default for first two symbols is a solid square and an open diamond. - `timeInc`: time increment for the x-axis. Default is 1/2 the shortest gap between any two distincttimes in the data. ## Returns a list containing `power`, `eff` (relative efficiency), and `approx.se` (approximate standard error of log odds ratio) for `popower`, or containing `n` and `eff` for `posamsize`. ## Author(s) Frank Harrell Department of Biostatistics Vanderbilt University School of Medicine fh@fharrell.com ## References Whitehead J (1993): Sample size calculations for ordered categorical data. Stat in Med 12:2257--2271. Julious SA, Campbell MJ (1996): Letter to the Editor. Stat in Med 15: 1065--1066. Shows accuracy of formula for binary response case. ## See Also `simRegOrd`, `bpower`, `cpower`, `impactPO` ## Examples ```r # For a study of back pain (none, mild, moderate, severe) here are the # expected proportions (averaged over 2 treatments) that will be in # each of the 4 categories: p <- c(.1,.2,.4,.3) popower(p, 1.2, 1000) # OR=1.2, total n=1000 posamsize(p, 1.2) popower(p, 1.2, 3148) # If p was the vector of probabilities for group 1, here's how to # compute the average over the two groups: # p2 <- pomodm(p=p, odds.ratio=1.2) # pavg <- (p + p2) / 2 # Compare power to test for proportions for binary case, # proportion of events in control group of 0.1 p <- 0.1; or <- 0.85; n <- 4000 popower(c(1 - p, p), or, n) # 0.338 bpower(p, odds.ratio=or, n=n) # 0.320 # Add more categories, starting with 0.1 in middle p <- c(.8, .1, .1) popower(p, or, n) # 0.543 p <- c(.7, .1, .1, .1) popower(p, or, n) # 0.67 # Continuous scale with final level have prob. 0.1 p <- c(rep(1 / n, 0.9 * n), 0.1) popower(p, or, n) # 0.843 # Compute the mean and median x after shifting the probability # distribution by an odds ratio under the proportional odds model x <- 1 : 5 p <- c(.05, .2, .2, .3, .25) # For comparison make up a sample that looks like this X <- rep(1 : 5, 20 * p) c(mean=mean(X), median=median(X)) pomodm(x, p, odds.ratio=1) # still have to figure out the right median pomodm(x, p, odds.ratio=0.5) # Show variation of odds ratios over possible cutoffs of Y even when PO # truly holds. Run 5 simulations for a total sample size of 300. # The two groups have 150 subjects each. s <- simPOcuts(300, nsim=5, odds.ratio=2, p=p) round(s, 2) # An ordinal outcome with levels a, b, c, d, e is measured at 3 times # Show the proportion of values in each outcome category stratified by # time. Then compute what the proportions would be had the proportions # at times 2 and 3 been the proportions at time 1 modified by two odds ratios set.seed(1) d <- expand.grid(time=1:3, reps=1:30) d$y <- sample(letters[1:5], nrow(d), replace=TRUE) propsPO(y ~ time, data=d, odds.ratio=function(time) c(1, 2, 4)[time]) # To show with plotly, save previous result as object p and then: # plotly::ggplotly(p, tooltip='label') # Add a stratification variable and don't consider an odds ratio d <- expand.grid(time=1:5, sex=c('female', 'male'), reps=1:30) d$y <- sample(letters[1:5], nrow(d), replace=TRUE) propsPO(y ~ time + sex, data=d) # may add nrow= or ncol= # Show all successive transition proportion matrices d <- expand.grid(id=1:30, time=1:10) d$state <- sample(LETTERS[1:4], nrow(d), replace=TRUE) propsTrans(state ~ time + id, data=d) pt1 <- data.frame(pt=1, day=0:3, status=c('well', 'well', 'sick', 'very sick')) pt2 <- data.frame(pt=2, day=c(1,2,4,6), status=c('sick', 'very sick', 'coma', 'death')) pt3 <- data.frame(pt=3, day=1:5, status=c('sick', 'very sick', 'sick', 'very sick', 'discharged')) pt4 <- data.frame(pt=4, day=c(1:4, 10), status=c('well', 'sick', 'very sick', 'well', 'discharged')) d <- rbind(pt1, pt2, pt3, pt4) d$status <- factor(d$status, c('discharged', 'well', 'sick', 'very sick', 'coma', 'death')) label(d$day) <- 'Day' require(ggplot2) multEventChart(status ~ day + pt, data=d, absorb=c('death', 'discharged'), colorTitle='Status', sortbylast=TRUE) + theme_classic() + theme(legend.position='bottom') ```

popower function