A movie to illustrate statistical concepts involved in the testing of one simple hypothesis against another. The example used is a random sample from a normal distribution whose variance is assumed to be known. The simple hypotheses relate to the value of the mean \(\mu\).
shypo(
mu0 = 0,
sd = 6,
eff = sd,
n = 10,
a = mu0 + eff/2,
target_alpha = 0.05,
target_beta = 0.1,
panel_plot = TRUE,
hscale = NA,
vscale = hscale,
delta_n = 1,
delta_a = sd/(10 * sqrt(n)),
delta_eff = sd,
delta_mu0 = 1,
delta_sd = 1
)A numeric scalar. The value of \(\mu\) under the null hypothesis H0 with which to start the movie.
A positive numeric scalar. The (common) standard deviation \(\sigma\) of the normal distributions of the data under the two hypotheses.
A numeric scalar. The effect size. The amount by which
the value of \(\mu\) under the alternative hypothesis is greater than
the value mu0 under the null hypothesis.
That is, mu1 = eff + mu0.
eff must be non-negative.
A positive integer scalar. The sample size with which to start the movie.
A numeric scalar. The critical value of the test with which to
start the movie. H0 is rejected if the sample mean is greater than
a.
A numeric scalar in (0,1). The target value of the
type I error to be achieved by setting a and/or n
if the user asks for this using a radio button.
A numeric scalar in (0,1). The target value of the
type II error to be achieved by setting a and/or n
if the user asks for this using a radio button.
A logical parameter that determines whether the plot
is placed inside the panel (TRUE) or in the standard graphics
window (FALSE). If the plot is to be placed inside the panel
then the tkrplot library is required.
Numeric scalars. Scaling parameters for the size
of the plot when panel_plot = TRUE. The default values are 1.4 on
Unix platforms and 2 on Windows platforms.
Numeric scalars. The
respective amounts by which the values of mu0, eff, a, n and
sd are increased (or decreased) after one click of the + (or -)
button in the parameter window.
Nothing is returned, only the animation is produced.
The movie is based on two plots.
The top plot shows the (normal)
probability density functions of the sample mean under the null
hypothesis H0 (mean mu0) and the alternative hypothesis H1
(mean mu1, where mu1 > mu0), with the values
of mu0 and mu1 indicated by vertical dashed lines.
H0 is rejected if the sample mean exceeds the critical value a,
which is indicated by a vertical black line.
The bottom plot shows how the probabilities of making a type I or type II
error (alpha and beta respectively) depend on the value of a,
by plotting these probabilities against a.
A parameter window enables the user to change the values of n,
a, mu0, eff = mu1 - mu0 or sd
by clicking the +/- buttons.
Radio buttons can be used either to:
set a to achieve the target type I error probability
target_alpha, based on the current value of n;
set a and (integer) n to achieve (or better) the
respective target type I and type II error probabilities of
target_alpha and target_beta.
If eff = 0 then a plot will be produced even though this case is
not practically meaningful. In the "set a and n to achieve target alpha
and beta" case, the plot will be the same as the case "set a and n by
hand" case.
# 1. Change a (for fixed n) to achieve alpha = 0.05
# 2. Change a and n to achieve alpha <= 0.05 and beta <= 0.1
shypo(mu0 = 0, eff = 5, n = 16, a = 2.3, delta_a = 0.01)