Summary of an Emax regression model — summary.emaxlogistic • emaxnls

Returns a tidy coefficient table for a fitted emaxnls or emaxlogistic model, combining parameter estimates, standard errors, test statistics, p-values, and confidence intervals.

Usage

# S3 method for class 'emaxlogistic'
summary(
  object,
  conf_level = 0.95,
  back_transform = FALSE,
  p_adjust = "none",
  simultaneous = FALSE,
  suppress_nonsensical = TRUE,
  ...
)

# S3 method for class 'emaxnls'
summary(
  object,
  conf_level = 0.95,
  back_transform = FALSE,
  p_adjust = "none",
  simultaneous = FALSE,
  suppress_nonsensical = TRUE,
  ...
)

Arguments

object: An emaxnls or emaxlogistic object
conf_level: Confidence level for interval estimates. Defaults to 0.95.
back_transform: Should logEC50 and logHill parameters be back-transformed to the concentration scale? If TRUE, these parameters are exponentiated and renamed to EC50 and Hill respectively, and their confidence intervals are transformed accordingly. Standard errors and test statistics on the back-transformed scale are not available and are set to NA.
p_adjust: Method for adjusting p-values for multiple comparisons, passed to stats::p.adjust(). Defaults to "none". Parameters with suppressed p-values (see suppress_nonsensical) are excluded from the adjustment set. This affects only the p_value column and is independent of simultaneous; see the "Multiplicity: p-value adjustment versus simultaneous intervals" section below.
simultaneous: If TRUE, compute simultaneous (joint) confidence intervals using the multivariate normal distribution via mvtnorm::qmvnorm(). This gives wider intervals that provide joint coverage at conf_level across all parameters simultaneously. Defaults to FALSE. This affects only the confidence-interval columns and is independent of p_adjust; see the "Multiplicity: p-value adjustment versus simultaneous intervals" section below.
suppress_nonsensical: If TRUE (the default), suppress the test statistic and p-value for logEC50_Intercept. The logEC50 intercept is estimated on the log-concentration scale, and testing H0: logEC50 = 0 is equivalent to testing whether EC50 equals 1 on the concentration scale — a threshold with no general pharmacometric meaning. The confidence interval for logEC50 is always reported regardless of this setting. Pass suppress_nonsensical = FALSE to restore the raw test results.
...: Ignored

Value

A tibble with one row per model parameter and columns for the estimate, standard error, test statistic, p-value, and confidence interval bounds. The column for the test statistic is named t_statistic for emaxnls models and z_statistic for emaxlogistic models. The return format is experimental and may change in future releases.

Details

Which tests are reported by default

Most parameters have a meaningful point null at zero:

Emax_Intercept: tests whether any exposure-response relationship exists.
logHill_Intercept: tests whether logHill = 0, i.e., whether the Hill parameter equals 1 on the concentration scale, which would mean the sigmoidal model reduces to a hyperbolic one.
E0_Intercept: tests whether the baseline response is zero. This is informative when the outcome is expressed as change from baseline, a common convention in pharmacometrics.
Covariate beta terms: test whether a given covariate has any effect on the corresponding structural parameter.

The one exception is logEC50_Intercept. The model is parameterized in terms of logEC50 (on the log-concentration scale), not EC50 directly, so the null H0: logEC50 = 0 corresponds to testing EC50 = 1 on the concentration scale — a value with no intrinsic pharmacometric meaning. By default, the test statistic and p-value for logEC50_Intercept are suppressed (set to NA), while the confidence interval for logEC50 is retained. To work with the EC50 on the concentration scale, use back_transform = TRUE.

Simultaneous intervals

When simultaneous = TRUE, a single critical value is derived from the joint multivariate normal distribution of the standardized parameter estimates. The resulting intervals have simultaneous coverage at conf_level and will be wider than the individual (pointwise) intervals.

Multiplicity: p-value adjustment versus simultaneous intervals

A model with several parameters raises a multiple-comparisons problem: the more quantities you inspect, the more likely at least one spurious result appears by chance. summary() offers two, deliberately separate, tools for this, and it is worth being clear about how they differ because they are easy to conflate.

p_adjust acts on the hypothesis tests. It takes the marginal (per-parameter) p-values and feeds them through stats::p.adjust(), which applies a sequential rule such as Holm or a Bonferroni scaling. These rules look only at the set of p-values; they do not use the estimated correlations between the parameters. Only the p_value column changes. The estimates, standard errors, test statistics, and confidence intervals are untouched.
simultaneous acts on the interval estimates. It replaces the per-parameter critical value with a single, larger critical value taken from the joint multivariate normal distribution of the estimates (via mvtnorm::qmvnorm()), which does use the correlation structure from vcov(). Only the ci_lower and ci_upper columns change. The p-values and test statistics are untouched.

The two arguments are fully independent: you may set either, both, or neither, and each does exactly the one thing described above. Neither argument modifies the other's output.

Why the adjusted p-values and the intervals may disagree

Because the two corrections use different machinery, they will not, in general, agree on which parameters are "significant". A parameter can have a Holm-adjusted p-value below 1 - conf_level while its simultaneous confidence interval still contains zero, or the reverse. This is not a bug. The familiar duality — "the 95% interval excludes zero if and only if the two-sided test rejects at the 5% level" — holds only for a single, unadjusted Wald comparison. It breaks as soon as a multiplicity correction enters, for two reasons:

the corrections answer different questions (a step-down rule on the p-values versus a joint critical region for the intervals), and
they use different information (p.adjust() ignores the parameter correlations that the simultaneous interval is built from).

A further, smaller source of discrepancy: unless the profile-likelihood computation falls back to Wald intervals, the default (pointwise) intervals are profile-likelihood based, whereas the reported test statistics and p-values are Wald quantities, so even without any adjustment the two need not correspond exactly in nonlinear models.

Which one to use

Pick the tool that matches the claim you want to make, and interpret its output on its own terms rather than cross-checking one against the other:

To report interval estimates that are jointly valid across all parameters, use simultaneous = TRUE.
To control the family-wise (or false-discovery) error rate of a set of hypothesis tests, choose a p_adjust method.

Setting both is legitimate, but the adjusted p-values and the simultaneous intervals are then two separate summaries of multiplicity, not two views of the same one, and should be read as such.

Examples

mod_c <- emax_nls(
  structural_model = rsp_1 ~ exp_1,
  covariate_model = list(E0 ~ cnt_a, Emax ~ 1, logEC50 ~ 1),
  data = emax_df,
  opts = emax_nls_options(max_time = 10)
)

# standard summary (logEC50_Intercept p-value suppressed by default)
summary(mod_c)
#> # A tibble: 4 × 7
#>   label             estimate std_error t_statistic    p_value ci_lower ci_upper
#>   <chr>                <dbl>     <dbl>       <dbl>      <dbl>    <dbl>    <dbl>
#> 1 E0_cnt_a             0.486    0.0116        42.1  3.63e-148    0.463    0.509
#> 2 E0_Intercept         5.05     0.0759        66.6  4.16e-217    4.91     5.20 
#> 3 Emax_Intercept       9.97     0.112         89.3  2.11e-264    9.75    10.2  
#> 4 logEC50_Intercept    8.27     0.0394        NA   NA            8.19     8.35 

# show all tests, including the logEC50 intercept
summary(mod_c, suppress_nonsensical = FALSE)
#> # A tibble: 4 × 7
#>   label             estimate std_error t_statistic   p_value ci_lower ci_upper
#>   <chr>                <dbl>     <dbl>       <dbl>     <dbl>    <dbl>    <dbl>
#> 1 E0_cnt_a             0.486    0.0116        42.1 3.63e-148    0.463    0.509
#> 2 E0_Intercept         5.05     0.0759        66.6 4.16e-217    4.91     5.20 
#> 3 Emax_Intercept       9.97     0.112         89.3 2.11e-264    9.75    10.2  
#> 4 logEC50_Intercept    8.27     0.0394       210.  0            8.19     8.35 

# Bonferroni-adjusted p-values
summary(mod_c, p_adjust = "bonferroni")
#> # A tibble: 4 × 7
#>   label             estimate std_error t_statistic    p_value ci_lower ci_upper
#>   <chr>                <dbl>     <dbl>       <dbl>      <dbl>    <dbl>    <dbl>
#> 1 E0_cnt_a             0.486    0.0116        42.1  1.09e-147    0.463    0.509
#> 2 E0_Intercept         5.05     0.0759        66.6  1.25e-216    4.91     5.20 
#> 3 Emax_Intercept       9.97     0.112         89.3  6.32e-264    9.75    10.2  
#> 4 logEC50_Intercept    8.27     0.0394        NA   NA            8.19     8.35 

# simultaneous confidence intervals
summary(mod_c, simultaneous = TRUE)
#> # A tibble: 4 × 7
#>   label             estimate std_error t_statistic    p_value ci_lower ci_upper
#>   <chr>                <dbl>     <dbl>       <dbl>      <dbl>    <dbl>    <dbl>
#> 1 E0_cnt_a             0.486    0.0116        42.1  3.63e-148    0.458    0.514
#> 2 E0_Intercept         5.05     0.0759        66.6  4.16e-217    4.87     5.24 
#> 3 Emax_Intercept       9.97     0.112         89.3  2.11e-264    9.70    10.2  
#> 4 logEC50_Intercept    8.27     0.0394        NA   NA            8.17     8.36 

# adjusted confidence level
summary(mod_c, conf_level = 0.99)
#> # A tibble: 4 × 7
#>   label             estimate std_error t_statistic    p_value ci_lower ci_upper
#>   <chr>                <dbl>     <dbl>       <dbl>      <dbl>    <dbl>    <dbl>
#> 1 E0_cnt_a             0.486    0.0116        42.1  3.63e-148    0.456    0.516
#> 2 E0_Intercept         5.05     0.0759        66.6  4.16e-217    4.86     5.25 
#> 3 Emax_Intercept       9.97     0.112         89.3  2.11e-264    9.68    10.3  
#> 4 logEC50_Intercept    8.27     0.0394        NA   NA            8.17     8.37 

# back-transform logEC50 and logHill to concentration scale
summary(mod_c, back_transform = TRUE)
#> # A tibble: 4 × 7
#>   label          estimate std_error t_statistic    p_value ci_lower ci_upper
#>   <chr>             <dbl>     <dbl>       <dbl>      <dbl>    <dbl>    <dbl>
#> 1 E0_cnt_a          0.486    0.0116        42.1  3.63e-148    0.463    0.509
#> 2 E0_Intercept      5.05     0.0759        66.6  4.16e-217    4.91     5.20 
#> 3 Emax_Intercept    9.97     0.112         89.3  2.11e-264    9.75    10.2  
#> 4 EC50_Intercept 3900.      NA             NA   NA         3608.    4211.   

# logistic emax equivalent
mod_b <- emax_logistic(
  structural_model = rsp_2 ~ exp_1,
  covariate_model = list(E0 ~ cnt_a, Emax ~ 1, logEC50 ~ 1),
  data = emax_df,
  opts = emax_logistic_options(max_time = 10)
)
summary(mod_b)
#> # A tibble: 4 × 7
#>   label             estimate std_error z_statistic   p_value ci_lower ci_upper
#>   <chr>                <dbl>     <dbl>       <dbl>     <dbl>    <dbl>    <dbl>
#> 1 E0_cnt_a             0.659    0.0800        8.24  1.79e-16    0.501    0.816
#> 2 E0_Intercept        -5.00     0.578        -8.64  5.43e-18   -6.14    -3.87 
#> 3 Emax_Intercept       8.12     2.27          3.58  3.45e- 4    5.08    17.6  
#> 4 logEC50_Intercept    9.78     0.518        NA    NA           8.89    11.0