The HII Galaxy Hubble Diagram Strongly Favors $R_{\rm h}=ct$ over
  $\Lambda$CDM
  We continue to build support for the proposal to use HII galaxies (HIIGx) and
giant extragalactic HII regions (GEHR) as standard candles to construct the
Hubble diagram at redshifts beyond the current reach of Type Ia supernovae.
Using a sample of 25 high-redshift HIIGx, 107 local HIIGx, and 24 GEHR, we
confirm that the correlation between the emission-line luminosity and
ionized-gas velocity dispersion is a viable luminosity indicator, and use it to
test and compare the standard model $\Lambda$CDM and the $R_{\rm h}=ct$
Universe by optimizing the parameters in each cosmology using a maximization of
the likelihood function. For the flat $\Lambda$CDM model, the best fit is
obtained with $\Omega_{\rm m}= 0.40_{-0.09}^{+0.09}$. However, statistical
tools, such as the Akaike (AIC), Kullback (KIC) and Bayes (BIC) Information
Criteria favor $R_{\rm h}=ct$ over the standard model with a likelihood of
$\approx 94.8\%-98.8\%$ versus only $\approx 1.2\%-5.2\%$. For $w$CDM (the
version of $\Lambda$CDM with a dark-energy equation of state $w_{\rm de}\equiv
p_{\rm de}/\rho_{\rm de}$ rather than $w_{\rm de}=w_{\Lambda}=-1$), a
statistically acceptable fit is realized with $\Omega_{\rm
m}=0.22_{-0.14}^{+0.16}$ and $w_{\rm de}= -0.51_{-0.25}^{+0.15}$ which,
however, are not fully consistent with their concordance values. In this case,
$w$CDM has two more free parameters than $R_{\rm h}=ct$, and is penalized more
heavily by these criteria. We find that $R_{\rm h}=ct$ is strongly favored over
$w$CDM with a likelihood of $\approx 92.9\%-99.6\%$ versus only $0.4\%-7.1\%$.
The current HIIGx sample is already large enough for the BIC to rule out
$\Lambda$CDM/$w$CDM in favor of $R_{\rm h}=ct$ at a confidence level
approaching $3\sigma$.
