ggpredict() computes estimated marginal means (predicted values) for the response, at the margin of specific values from certain model terms. ggeffect() computes marginal effects by internally calling Effect, while ggemmeans() computes marginal effects by internally calling emmeans. The result is returned as consistent data frame.

ggeffect(model, terms, ci.lvl = 0.95, ...)

ggemmeans(
  model,
  terms,
  ci.lvl = 0.95,
  type = "fe",
  typical = "mean",
  condition = NULL,
  back.transform = TRUE,
  ...
)

ggpredict(
  model,
  terms,
  ci.lvl = 0.95,
  type = "fe",
  typical = "mean",
  condition = NULL,
  back.transform = TRUE,
  ppd = FALSE,
  vcov.fun = NULL,
  vcov.type = NULL,
  vcov.args = NULL,
  interval = "confidence",
  ...
)

Arguments

model

A fitted model object, or a list of model objects. Any model that supports common methods like predict(), family() or model.frame() should work. For ggeffect(), any model that is supported by effects should work, and for ggemmeans(), all models supported by emmeans should work.

terms

Character vector (or a formula) with the names of those terms from model, for which marginal effects should be displayed. At least one term is required to calculate effects for certain terms, maximum length is four terms, where the second to fourth term indicate the groups, i.e. predictions of first term are grouped at the values or levels of the remaining terms. If terms is missing or NULL, marginal effects for each model term are calculated. It is also possible to define specific values for terms, at which marginal effects should be calculated (see 'Details'). All remaining covariates that are not specified in terms are held constant (see 'Details'). See also arguments condition and typical.

ci.lvl

Numeric, the level of the confidence intervals. For ggpredict(), use ci.lvl = NA, if confidence intervals should not be calculated (for instance, due to computation time). Typically, confidence intervals based on the standard errors as returned by the predict() function are returned, assuming normal distribution (i.e. +/- 1.96 * SE). See introduction of this vignette for more details.

...

For ggpredict(), further arguments passed down to predict(); for ggeffect(), further arguments passed down to Effect; and for ggemmeans(), further arguments passed down to emmeans. If type = "sim", ... may also be used to set the number of simulation, e.g. nsim = 500.

type

Character, only applies for survival models, mixed effects models and/or models with zero-inflation. Note: For brmsfit-models with zero-inflation component, there is no type = "fe.zi" nor type = "re.zi"; predicted values for MixMod-models from GLMMadaptive with zero-inflation component always condition on the zero-inflation part of the model (see 'Details').

"fe"

Predicted values are conditioned on the fixed effects or conditional model only (for mixed models: predicted values are on the population-level and confidence intervals are returned). For instance, for models fitted with zeroinfl from pscl, this would return the predicted mean from the count component (without zero-inflation). For models with zero-inflation component, this type calls predict(..., type = "link") (however, predicted values are back-transformed to the response scale).

"re"

This only applies to mixed models, and type = "re" does not condition on the zero-inflation component of the model. type = "re" still returns population-level predictions, however, unlike type = "fe", intervals also consider the uncertainty in the variance parameters (the mean random effect variance, see Johnson et al. 2014 for details) and hence can be considered as prediction intervals. For models with zero-inflation component, this type calls predict(..., type = "link") (however, predicted values are back-transformed to the response scale).

To get predicted values for each level of the random effects groups, add the name of the related random effect term to the terms-argument (for more details, see this vignette).

"fe.zi"

Predicted values are conditioned on the fixed effects and the zero-inflation component. For instance, for models fitted with zeroinfl from pscl, this would return the predicted response (mu*(1-p)) and for glmmTMB, this would return the expected value mu*(1-p) without conditioning on random effects (i.e. random effect variances are not taken into account for the confidence intervals). For models with zero-inflation component, this type calls predict(..., type = "response"). See 'Details'.

"re.zi"

Predicted values are conditioned on the zero-inflation component and take the random effects uncertainty into account. For models fitted with glmmTMB(), hurdle() or zeroinfl(), this would return the expected value mu*(1-p). For glmmTMB, prediction intervals also consider the uncertainty in the random effects variances. This type calls predict(..., type = "response"). See 'Details'.

"sim"

Predicted values and confidence resp. prediction intervals are based on simulations, i.e. calls to simulate(). This type of prediction takes all model uncertainty into account, including random effects variances. Currently supported models are glmmTMB and merMod. See ... for details on number of simulations.

"surv" and "cumhaz"

Applies only to coxph-objects from the survial-package and calculates the survival probability or the cumulative hazard of an event.

typical

Character vector, naming the function to be applied to the covariates over which the effect is "averaged". The default is "mean". See typical_value for options.

condition

Named character vector, which indicates covariates that should be held constant at specific values. Unlike typical, which applies a function to the covariates to determine the value that is used to hold these covariates constant, condition can be used to define exact values, for instance condition = c(covariate1 = 20, covariate2 = 5). See 'Examples'.

back.transform

Logical, if TRUE (the default), predicted values for log- or log-log transformend responses will be back-transformed to original response-scale.

ppd

Logical, if TRUE, predictions for Stan-models are based on the posterior predictive distribution (posterior_predict). If FALSE (the default), predictions are based on posterior draws of the linear predictor (posterior_linpred).

vcov.fun

String, indicating the name of the vcov*()-function from the sandwich-package, e.g. vcov.fun = "vcovCL", which is used to compute robust standard errors for predictions. If NULL, standard errors (and confidence intervals) for predictions are based on the standard errors as returned by the predict()-function. Note that probably not all model objects that work with ggpredict() are also supported by the sandwich-package.

vcov.type

Character vector, specifying the estimation type for the robust covariance matrix estimation (see vcovHC for details).

vcov.args

List of named vectors, used as additional arguments that are passed down to vcov.fun.

interval

Type of interval calculation, can either be "confidence" (default) or "prediction". May be abbreviated. Unlike confidence intervals, prediction intervals include the residual variance (sigma^2). This argument is ignored for mixed models, as interval = "prediction" is equivalent to type = "re" (and interval = "confidence" is equivalent to type = "fe"). Note that prediction intervals are not available for all models.

Value

A data frame (with ggeffects class attribute) with consistent data columns:

x

the values of the first term in terms, used as x-position in plots.

predicted

the predicted values of the response, used as y-position in plots.

std.error

the standard error of the predictions.

conf.low

the lower bound of the confidence interval for the predicted values.

conf.high

the upper bound of the confidence interval for the predicted values.

group

the grouping level from the second term in terms, used as grouping-aesthetics in plots.

facet

the grouping level from the third term in terms, used to indicate facets in plots.

The estimated marginal means (predicted values) are always on the response scale!

For proportional odds logistic regression (see polr) resp. cumulative link models (e.g., see clm), an additional column response.level is returned, which indicates the grouping of predictions based on the level of the model's response.

Note that for convenience reasons, the columns for the intervals are always named conf.low and conf.high, even though for Bayesian models credible or highest posterior density intervals are returned.

Details

Supported Models

A list of supported models can be found at https://github.com/strengejacke/ggeffects. Support for models varies by function, i.e. although ggpredict(), ggemmeans() and ggeffect() support most models, some models are only supported exclusively by one of the three functions.

Difference between ggpredict() and ggeffect() or ggemmeans()

ggpredict() calls predict(), while ggeffect() calls Effect and ggemmeans() calls emmeans to compute marginal effects. Therefore, ggpredict() and ggeffect() resp. ggemmeans() differ in how factors are held constant: ggpredict() uses the reference level, while ggeffect() and ggemmeans() compute a kind of "average" value, which represents the proportions of each factor's category. Use condition to set a specific level for factors in ggemmeans(), so factors are not averaged over their categories, but held constant at a given level.

Marginal Effects at Specific Values

Specific values of model terms can be specified via the terms-argument. Indicating levels in square brackets allows for selecting only specific groups or values resp. value ranges. Term name and the start of the levels in brackets must be separated by a whitespace character, e.g. terms = c("age", "education [1,3]"). Numeric ranges, separated with colon, are also allowed: terms = c("education", "age [30:60]").

The terms-argument also supports the same shortcuts as the values-argument in values_at(). So terms = "age [meansd]" would return predictions for the values one standard deviation below the mean age, the mean age and one SD above the mean age. terms = "age [quart2]" would calculate predictions at the value of the lower, median and upper quartile of age.

Furthermore, it is possible to specify a function name. Values for predictions will then be transformed, e.g. terms = "income [exp]". This is useful when model predictors were transformed for fitting the model and should be back-transformed to the original scale for predictions. It is also possible to define own functions (see this vignette).

You can take a random sample of any size with sample=n, e.g terms = "income [sample=8]", which will sample eight values from all possible values of the variable income. This option is especially useful for plotting marginal effects at certain levels of random effects group levels, where the group factor has many levels that can be completely plotted. For more details, see this vignette.

Finally, numeric vectors for which no specific values are given, a "pretty range" is calculated (see pretty_range), to avoid memory allocation problems for vectors with many unique values. If a numeric vector is specified as second or third term (i.e. if this vector represents a grouping structure), representative values (see values_at) are chosen (unless other values are specified). If all values for a numeric vector should be used to compute predictions, you may use e.g. terms = "age [all]". See also package vignettes.

To create a pretty range that should be smaller or larger than the default range (i.e. if no specific values would be given), use the n-tag, e.g. terms="age [n=5]" or terms="age [n=12]". Larger values for n return a larger range of predicted values.

Holding covariates at constant values

For ggpredict(), expand.grid() is called on all unique combinations of model.frame(model)[, terms] and used as newdata-argument for predict(). In this case, all remaining covariates that are not specified in terms are held constant: Numeric values are set to the mean (unless changed with the condition or typical-argument), factors are set to their reference level (may also be changed with condition) and character vectors to their mode (most common element).

ggeffect() and ggemmeans(), by default, set remaining numeric covariates to their mean value, while for factors, a kind of "average" value, which represents the proportions of each factor's category, is used. For ggemmeans(), use condition to set a specific level for factors so that these are not averaged over their categories, but held constant at the given level.

Bayesian Regression Models

ggpredict() also works with Stan-models from the rstanarm or brms-package. The predicted values are the median value of all drawn posterior samples. The confidence intervals for Stan-models are Bayesian predictive intervals. By default (i.e. ppd = FALSE), the predictions are based on posterior_linpred and hence have some limitations: the uncertainty of the error term is not taken into account. The recommendation is to use the posterior predictive distribution (posterior_predict).

Zero-Inflated and Zero-Inflated Mixed Models with brms

Models of class brmsfit always condition on the zero-inflation component, if the model has such a component. Hence, there is no type = "fe.zi" nor type = "re.zi" for brmsfit-models, because predictions are based on draws of the posterior distribution, which already account for the zero-inflation part of the model.

Zero-Inflated and Zero-Inflated Mixed Models with glmmTMB

If model is of class glmmTMB, hurdle, zeroinfl or zerotrunc, simulations from a multivariate normal distribution (see mvrnorm) are drawn to calculate mu*(1-p). Confidence intervals are then based on quantiles of these results. For type = "re.zi", prediction intervals also take the uncertainty in the random-effect paramters into account (see also Brooks et al. 2017, pp.391-392 for details).

An alternative for models fitted with glmmTMB that take all model uncertainties into account are simulations based on simulate(), which is used when type = "sim" (see Brooks et al. 2017, pp.392-393 for details).

MixMod-models from GLMMadaptive

Predicted values for the fixed effects component (type = "fe" or type = "fe.zi") are based on predict(..., type = "mean_subject"), while predicted values for random effects components (type = "re" or type = "re.zi") are calculated with predict(..., type = "subject_specific") (see ?GLMMadaptive::predict.MixMod for details). The latter option requires the response variable to be defined in the newdata-argument of predict(), which will be set to its typical value (see typical_value).

Note

Multinomial Models

polr-, clm-models, or more generally speaking, models with ordinal or multinominal outcomes, have an additional column response.level, which indicates with which level of the response variable the predicted values are associated.

Printing Results

The print()-method gives a clean output (especially for predictions by groups), and indicates at which values covariates were held constant. Furthermore, the print()-method has the arguments digits and n to control number of decimals and lines to be printed, and an argument x.lab to print factor-levels instead of numeric values if x is a factor.

Limitations

The support for some models, for example from package MCMCglmm, is rather experimental and may fail for certain models. If you encounter any errors, please file an issue at https://github.com/strengejacke/ggeffects/issues.

References

  • Brooks ME, Kristensen K, Benthem KJ van, Magnusson A, Berg CW, Nielsen A, et al. glmmTMB Balances Speed and Flexibility Among Packages for Zero-inflated Generalized Linear Mixed Modeling. The R Journal. 2017;9: 378-400.

  • Johnson PC, O'Hara RB. 2014. Extension of Nakagawa & Schielzeth's R2GLMM to random slopes models. Methods Ecol Evol, 5: 944-946. (doi: 10.1111/2041-210X.12225 )

Examples

library(sjlabelled) data(efc) fit <- lm(barthtot ~ c12hour + neg_c_7 + c161sex + c172code, data = efc) ggpredict(fit, terms = "c12hour")
#> #> # Predicted values of Total score BARTHEL INDEX #> # x = average number of hours of care per week #> #> x predicted std.error conf.low conf.high #> 0 75.444 1.116 73.257 77.630 #> 20 70.378 0.925 68.564 72.191 #> 45 64.045 0.843 62.393 65.697 #> 65 58.979 0.930 57.157 60.802 #> 85 53.913 1.122 51.713 56.113 #> 105 48.847 1.377 46.148 51.546 #> 125 43.781 1.665 40.517 47.045 #> 170 32.382 2.373 27.732 37.033 #> #> Adjusted for: #> * neg_c_7 = 11.84 #> * c161sex = 1.76 #> * c172code = 1.97 #>
ggpredict(fit, terms = c("c12hour", "c172code"))
#> #> # Predicted values of Total score BARTHEL INDEX #> # x = average number of hours of care per week #> #> # c172code = low level of education #> x predicted std.error conf.low conf.high #> 0 74.746 1.776 71.266 78.227 #> 30 67.147 1.587 64.037 70.257 #> 55 60.815 1.550 57.777 63.853 #> 85 53.216 1.662 49.958 56.473 #> 115 45.617 1.914 41.865 49.368 #> 170 31.685 2.593 26.602 36.768 #> #> # c172code = intermediate level of education #> x predicted std.error conf.low conf.high #> 0 75.465 1.114 73.282 77.647 #> 30 67.866 0.868 66.165 69.566 #> 55 61.533 0.872 59.824 63.243 #> 85 53.934 1.126 51.727 56.141 #> 115 46.335 1.522 43.352 49.318 #> 170 32.404 2.377 27.745 37.062 #> #> # c172code = high level of education #> x predicted std.error conf.low conf.high #> 0 76.183 1.718 72.816 79.550 #> 30 68.584 1.616 65.417 71.751 #> 55 62.252 1.656 59.006 65.497 #> 85 54.653 1.843 51.040 58.265 #> 115 47.053 2.143 42.853 51.254 #> 170 33.122 2.863 27.511 38.733 #> #> Adjusted for: #> * neg_c_7 = 11.84 #> * c161sex = 1.76 #>
ggpredict(fit, terms = c("c12hour", "c172code", "c161sex"))
#> #> # Predicted values of Total score BARTHEL INDEX #> # x = average number of hours of care per week #> #> # c172code = low level of education #> # c161sex = [1] Male #> x predicted std.error conf.low conf.high #> 0 73.954 2.347 69.354 78.554 #> 45 62.556 2.208 58.228 66.883 #> 85 52.424 2.310 47.896 56.951 #> 170 30.893 3.085 24.847 36.939 #> #> # c172code = intermediate level of education #> # c161sex = [1] Male #> x predicted std.error conf.low conf.high #> 0 74.673 1.845 71.055 78.290 #> 45 63.274 1.730 59.883 66.665 #> 85 53.142 1.911 49.397 56.887 #> 170 31.611 2.872 25.982 37.241 #> #> # c172code = high level of education #> # c161sex = [1] Male #> x predicted std.error conf.low conf.high #> 0 75.391 2.220 71.040 79.741 #> 45 63.992 2.176 59.727 68.258 #> 85 53.860 2.364 49.226 58.494 #> 170 32.330 3.257 25.946 38.713 #> #> # c172code = low level of education #> # c161sex = [2] Female #> x predicted std.error conf.low conf.high #> 0 74.996 1.831 71.406 78.585 #> 45 63.597 1.603 60.456 66.738 #> 85 53.465 1.702 50.130 56.800 #> 170 31.934 2.606 26.827 37.042 #> #> # c172code = intermediate level of education #> # c161sex = [2] Female #> x predicted std.error conf.low conf.high #> 0 75.714 1.225 73.313 78.115 #> 45 64.315 0.968 62.418 66.213 #> 85 54.183 1.209 51.815 56.552 #> 170 32.653 2.403 27.943 37.362 #> #> # c172code = high level of education #> # c161sex = [2] Female #> x predicted std.error conf.low conf.high #> 0 76.432 1.809 72.887 79.977 #> 45 65.034 1.712 61.679 68.388 #> 85 54.902 1.910 51.158 58.646 #> 170 33.371 2.895 27.697 39.045 #> #> Adjusted for: #> * neg_c_7 = 11.84 #>
# specified as formula ggpredict(fit, terms = ~ c12hour + c172code + c161sex)
#> #> # Predicted values of Total score BARTHEL INDEX #> # x = average number of hours of care per week #> #> # c172code = low level of education #> # c161sex = [1] Male #> x predicted std.error conf.low conf.high #> 0 73.954 2.347 69.354 78.554 #> 45 62.556 2.208 58.228 66.883 #> 85 52.424 2.310 47.896 56.951 #> 170 30.893 3.085 24.847 36.939 #> #> # c172code = intermediate level of education #> # c161sex = [1] Male #> x predicted std.error conf.low conf.high #> 0 74.673 1.845 71.055 78.290 #> 45 63.274 1.730 59.883 66.665 #> 85 53.142 1.911 49.397 56.887 #> 170 31.611 2.872 25.982 37.241 #> #> # c172code = high level of education #> # c161sex = [1] Male #> x predicted std.error conf.low conf.high #> 0 75.391 2.220 71.040 79.741 #> 45 63.992 2.176 59.727 68.258 #> 85 53.860 2.364 49.226 58.494 #> 170 32.330 3.257 25.946 38.713 #> #> # c172code = low level of education #> # c161sex = [2] Female #> x predicted std.error conf.low conf.high #> 0 74.996 1.831 71.406 78.585 #> 45 63.597 1.603 60.456 66.738 #> 85 53.465 1.702 50.130 56.800 #> 170 31.934 2.606 26.827 37.042 #> #> # c172code = intermediate level of education #> # c161sex = [2] Female #> x predicted std.error conf.low conf.high #> 0 75.714 1.225 73.313 78.115 #> 45 64.315 0.968 62.418 66.213 #> 85 54.183 1.209 51.815 56.552 #> 170 32.653 2.403 27.943 37.362 #> #> # c172code = high level of education #> # c161sex = [2] Female #> x predicted std.error conf.low conf.high #> 0 76.432 1.809 72.887 79.977 #> 45 65.034 1.712 61.679 68.388 #> 85 54.902 1.910 51.158 58.646 #> 170 33.371 2.895 27.697 39.045 #> #> Adjusted for: #> * neg_c_7 = 11.84 #>
# only range of 40 to 60 for variable 'c12hour' ggpredict(fit, terms = "c12hour [40:60]")
#> #> # Predicted values of Total score BARTHEL INDEX #> # x = average number of hours of care per week #> #> x predicted std.error conf.low conf.high #> 40 65.312 0.842 63.661 66.962 #> 43 64.552 0.842 62.902 66.201 #> 45 64.045 0.843 62.393 65.697 #> 47 63.538 0.846 61.881 65.196 #> 50 62.779 0.852 61.108 64.449 #> 53 62.019 0.862 60.329 63.708 #> 55 61.512 0.870 59.806 63.218 #> 60 60.246 0.896 58.489 62.002 #> #> Adjusted for: #> * neg_c_7 = 11.84 #> * c161sex = 1.76 #> * c172code = 1.97 #>
# using "summary()" shows that covariate "neg_c_7" is held # constant at a value of 11.84 (its mean value). To use a # different value, use "condition" ggpredict(fit, terms = "c12hour [40:60]", condition = c(neg_c_7 = 20))
#> #> # Predicted values of Total score BARTHEL INDEX #> # x = average number of hours of care per week #> #> x predicted std.error conf.low conf.high #> 40 46.562 2.030 42.582 50.541 #> 43 45.802 2.018 41.847 49.757 #> 45 45.295 2.010 41.355 49.235 #> 47 44.789 2.003 40.862 48.715 #> 50 44.029 1.994 40.121 47.936 #> 53 43.269 1.986 39.377 47.161 #> 55 42.762 1.981 38.879 46.645 #> 60 41.496 1.972 37.631 45.360 #> #> Adjusted for: #> * c161sex = 1.76 #> * c172code = 1.97 #>
# to plot ggeffects-objects, you can use the 'plot()'-function. # the following examples show how to build your ggplot by hand. # \donttest{ # plot predicted values, remaining covariates held constant library(ggplot2) mydf <- ggpredict(fit, terms = "c12hour") ggplot(mydf, aes(x, predicted)) + geom_line() + geom_ribbon(aes(ymin = conf.low, ymax = conf.high), alpha = .1)
# three variables, so we can use facets and groups mydf <- ggpredict(fit, terms = c("c12hour", "c161sex", "c172code")) ggplot(mydf, aes(x = x, y = predicted, colour = group)) + stat_smooth(method = "lm", se = FALSE) + facet_wrap(~facet, ncol = 2)
# select specific levels for grouping terms mydf <- ggpredict(fit, terms = c("c12hour", "c172code [1,3]", "c161sex")) ggplot(mydf, aes(x = x, y = predicted, colour = group)) + stat_smooth(method = "lm", se = FALSE) + facet_wrap(~facet) + labs( y = get_y_title(mydf), x = get_x_title(mydf), colour = get_legend_title(mydf) )
# level indication also works for factors with non-numeric levels # and in combination with numeric levels for other variables data(efc) efc$c172code <- as_label(efc$c172code) fit <- lm(barthtot ~ c12hour + neg_c_7 + c161sex + c172code, data = efc) ggpredict(fit, terms = c("c12hour", "c172code [low level of education, high level of education]", "c161sex [1]"))
#> #> # Predicted values of Total score BARTHEL INDEX #> # x = average number of hours of care per week #> #> # c172code = low level of education #> x predicted std.error conf.low conf.high #> 0 72.813 2.501 67.912 77.714 #> 30 65.216 2.386 60.540 69.892 #> 55 58.886 2.375 54.231 63.540 #> 85 51.289 2.464 46.460 56.118 #> 115 43.692 2.654 38.490 48.895 #> 170 29.765 3.200 23.493 36.037 #> #> # c172code = high level of education #> x predicted std.error conf.low conf.high #> 0 74.030 2.447 69.233 78.826 #> 30 66.433 2.392 61.744 71.122 #> 55 60.102 2.432 55.336 64.869 #> 85 52.506 2.577 47.454 57.557 #> 115 44.909 2.813 39.396 50.421 #> 170 30.982 3.412 24.293 37.670 #> #> Adjusted for: #> * neg_c_7 = 11.84 #>
# use categorical value on x-axis, use axis-labels, add error bars dat <- ggpredict(fit, terms = c("c172code", "c161sex")) ggplot(dat, aes(x, predicted, colour = group)) + geom_point(position = position_dodge(.1)) + geom_errorbar( aes(ymin = conf.low, ymax = conf.high), position = position_dodge(.1) ) + scale_x_discrete(breaks = 1:3, labels = get_x_labels(dat))
# 3-way-interaction with 2 continuous variables data(efc) # make categorical efc$c161sex <- as_factor(efc$c161sex) fit <- lm(neg_c_7 ~ c12hour * barthtot * c161sex, data = efc) # select only levels 30, 50 and 70 from continuous variable Barthel-Index dat <- ggpredict(fit, terms = c("c12hour", "barthtot [30,50,70]", "c161sex")) ggplot(dat, aes(x = x, y = predicted, colour = group)) + stat_smooth(method = "lm", se = FALSE, fullrange = TRUE) + facet_wrap(~facet) + labs( colour = get_legend_title(dat), x = get_x_title(dat), y = get_y_title(dat), title = get_title(dat) )
# or with ggeffects' plot-method plot(dat, ci = FALSE)# }
# marginal effects for polynomial terms data(efc) fit <- glm( tot_sc_e ~ c12hour + e42dep + e17age + I(e17age^2) + I(e17age^3), data = efc, family = poisson() ) ggeffect(fit, terms = "e17age")
#> #> # Predicted values of Services for elderly #> # x = elder' age #> #> x predicted std.error conf.low conf.high #> 64 1.369 0.140 1.040 1.801 #> 70 0.942 0.059 0.840 1.057 #> 74 0.899 0.059 0.800 1.010 #> 78 0.939 0.050 0.852 1.036 #> 84 1.042 0.052 0.941 1.153 #> 90 1.007 0.066 0.884 1.146 #> 94 0.824 0.121 0.650 1.044 #> 104 0.172 0.697 0.044 0.674 #>