Convert Between an F Value and a Chi Square Value

Converts an observed F value into a chi square value, and a chi square value into an F value. Both functions return the converted statistic itself, a single number, not a p-value.

Two conversions are available, selected by df_denominator.

Scaling (the default). With df_denominator = Inf, convert_F_chisq() returns df_numerator * F_value and convert_chisq_F() returns chi_square / df. This is the standard conversion between the two test statistics and needs no denominator degrees of freedom.
Probability matching. With a finite df_denominator, convert_F_chisq() returns the chi square value that has the same upper-tail probability (the same p-value) as the F value, and convert_chisq_F() returns the F value with the same upper-tail probability as the chi square value.

Usage

convert_F_chisq(F_value, df_numerator, df_denominator = Inf)

convert_chisq_F(chi_square, df, df_denominator = Inf)

Arguments

F_value: Observed F value. Must be nonnegative.
df_numerator: Numerator degrees of freedom of the F, which is also the degrees of freedom of the chi square.
df_denominator: Denominator degrees of freedom of the F, the degrees of freedom on which the error variance is estimated. The default, Inf, treats the error variance as known and gives the scaling conversion ($\chi^2 = \nu_1 F$); a finite value gives the probability-matching conversion. See Details.
chi_square: Observed chi square value. Must be nonnegative.
df: Degrees of freedom of the chi square, which becomes the numerator degrees of freedom of the F.

Value

A 1-row data.frame with columns term and value. The term is "chi_square_from_F" for convert_F_chisq and "F_from_chi_square" for convert_chisq_F, and value is the converted statistic (a chi square value or an F value, respectively; never a p-value).

Details

Why there are two conversions. An F statistic is the ratio of two independent chi squares, each divided by its degrees of freedom, $$F(\nu_1, \nu_2) = \frac{\chi^2_{\nu_1}/\nu_1}{\chi^2_{\nu_2}/\nu_2},$$ where the denominator is the estimated error variance scaled to have a mean of 1. A chi square is what the numerator becomes when that error variance is known rather than estimated. This is the entire difference between the two, and it is why the conversion depends on how the error variance is treated.

Scaling: df_denominator = Inf. As the denominator degrees of freedom grow, the estimated error variance converges to the true one and $\nu_1 F \to \chi^2(\nu_1)$. The default therefore treats the error variance as known and returns exactly $$\chi^2 = \nu_1 \, F, \qquad F = \chi^2 / \nu_1,$$ with $\nu_1$ the numerator degrees of freedom (df_numerator, which is also the degrees of freedom of the chi square). This is the value an F table prints in its infinite-denominator row, and it is the usual conversion between a Wald F and a Wald chi square. It involves no probabilities and needs no df_denominator.

Probability matching: finite df_denominator. When the error variance is estimated on $\nu_2$ degrees of freedom, the scaling above runs high, because $\nu_1 F$ is more dispersed than $\chi^2(\nu_1)$. Supplying df_denominator returns instead the chi square value at the same upper-tail probability. Writing pf and qchisq for R's distribution and quantile functions, the computation is exactly

p <- pf(F_value, df_numerator, df_denominator, lower.tail = FALSE)
chi_square <- qchisq(p, df_numerator, lower.tail = FALSE)

and convert_chisq_F() composes the same two functions in the other order. The upper tail is used so the p-value is represented accurately for large statistics (the lower-tail probability rounds to 1 in double precision by about $F = 500$ at small $\nu_2$, which would send qchisq() to infinity; the upper-tail value stays accurate past $F = 10^{20}$). No logarithms are involved and the returned value is the chi square itself, not the p-value used to find it. The map is strictly increasing and therefore one to one, and convert_chisq_F() is its exact inverse.

How much the two differ. At $\nu_1 = 3$ and $F = 2.75$, probability matching gives $\chi^2 = 6.290$ at $\nu_2 = 10$, $7.711$ at $\nu_2 = 50$, and $8.220$ at $\nu_2 = 1000$, approaching the scaling value $\nu_1 F = 8.25$ only as $\nu_2$ grows. Scaling and probability matching agree in the limit and diverge as $\nu_2$ shrinks; for a small $\nu_2$, the scaled value understates the p-value (with $\nu_1 = 3, \nu_2 = 5$, a result whose true p is .05 reads as .001 if $\nu_1 F$ is referred to $\chi^2$).

Noncentrality. Both conversions are defined by the central distributions and preserve the p-value; neither transports a noncentrality parameter (a noncentral F is not carried to a noncentral chi square with the same $\lambda$, except in the $\nu_2 \to \infty$ limit). For noncentral work use conf_limits_ncf and conf_limits_nc_chisq.

Special case. With $\nu_1 = 1$ this is the squared form of the relation between t and z, since $F(1, \nu) = t(\nu)^2$ and $\chi^2(1) = z^2$.

References

Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). Continuous univariate distributions (2nd ed., Vol. 2). Wiley.

Maxwell, S. E., Delaney, H. D., & Kelley, K. (2027). Designing experiments and analyzing data: A model comparison perspective (4th ed.). Routledge.

Author

Ken Kelley kkelley@nd.edu

Examples

# Scaling (the default): chi square = df_numerator * F.
convert_F_chisq(2.75, df_numerator = 3)
#>  term              value
#>  chi_square_from_F 8.25 

# The inverse: F = chi square / df.
convert_chisq_F(8.25, df = 3)
#>  term              value
#>  F_from_chi_square 2.75 

# Probability matching at 3 and 50 degrees of freedom: the chi square
# with the same p-value as the F.
convert_F_chisq(2.75, df_numerator = 3, df_denominator = 50)
#>  term              value
#>  chi_square_from_F 7.71 

# The p-value is preserved, which is what probability matching means.
f <- 2.75
x <- convert_F_chisq(f, df_numerator = 3, df_denominator = 50)$value
c(p_from_F = pf(f, 3, 50, lower.tail = FALSE),
  p_from_chi_square = pchisq(x, 3, lower.tail = FALSE))
#>          p_from_F p_from_chi_square 
#>        0.05238213        0.05238213