Provides the Critical Value(s) for an F Distribution

Usage

cv_f(
  alpha_level,
  df_numerator,
  df_denominator,
  alternative = "greater",
  alpha_lower,
  alpha_upper,
  ncp = 0,
  verbose = TRUE
)

Arguments

alpha_level: Type I error rate (i.e., the false positive rate).
df_numerator: The numerator degrees of freedom (a positive number). In a model comparison this is the difference in the number of parameters between the two models.
df_denominator: The denominator (error) degrees of freedom (a positive number).
alternative: The type of alternative hypothesis of interest. The default, "greater", puts the whole of alpha_level in the upper tail, which is how the F distribution is used to test a model comparison (see Details).
alpha_lower: The error rate in the lower tail of the distribution.
alpha_upper: The error rate in the upper tail of the distribution.
ncp: The noncentral parameter (if zero, the default, it is the central F distribution).
verbose: Provides extra information about areas under the curve.

Value

Returns the critical value(s), based on the input specifications, in a output style (a data.frame with a row for the lower and the upper critical value, following the format used by cv_t).

Details

Unlike the t and z distributions, the F distribution is not symmetric and takes only non-negative values, and the usual test of a model comparison is one-sided: a restricted model fits worse than a full model, so evidence against the restriction shows up as a large F, never a small one. That is why alternative defaults to "greater" here whereas it defaults to "not_equal" in cv_t. Maxwell, Delaney, and Kelley (2027) tabulate these upper-tail values in their Appendix Table A.2.

Both tails remain available for the situations that need them, such as an interval for a ratio of variances, either by setting alternative = "not_equal" or by giving alpha_lower and alpha_upper directly. When a tail is given zero area its critical value is the boundary of the support, so lower_cv is 0 under the default.

A noncentral parameter can be supplied, which is what a power analysis needs, though it would not be used for a standard null hypothesis significance test. See conf_limits_ncf for confidence limits on the noncentral parameter itself.

References

Maxwell, S. E., Delaney, H. D., & Kelley, K. (2027). Designing experiments and analyzing data: A model comparison perspective (4th ed.). Routledge. (See Chapter 3, where the F test of a model comparison is developed; Appendix Table A.2 reports these critical values.)

Author

Ken Kelley kkelley@nd.edu

Examples

# The critical value for a model comparison with 3 numerator and 20
# denominator degrees of freedom, at the .05 level.
cv_f(alpha_level = .05, df_numerator = 3, df_denominator = 20)
#>  term     value area_less area_greater
#>  lower_cv 0     0         1           
#>  upper_cv 3.1   0.95      0.05        

# An omnibus test of four groups with 24 participants: a - 1 = 3 and
# N - a = 20 degrees of freedom; simple output.
cv_f(alpha_level = .05, df_numerator = 3, df_denominator = 20, verbose = FALSE)
#>  term     value
#>  lower_cv 0    
#>  upper_cv 3.1  

# Both tails, as an interval for a ratio of variances would need.
cv_f(alpha_level = .05, df_numerator = 3, df_denominator = 20,
     alternative = "not_equal")
#>  term     value  area_less area_greater
#>  lower_cv 0.0706 0.025     0.975       
#>  upper_cv 3.86   0.975     0.025