Confusion with Chi Squared interpretation I'm getting confused interpreting the chi squared distribution. I have a number of chi squared values between 10 and ~100 for a number of parameter estimations fitting data to a model and I have 25 degrees of freedom. My null hypothesis is that the model tested is the true model. How do I reject the null hypothesis? 
Looking up the table of critical values (https://www.medcalc.org/manual/chi-square-table.php ) tells me that for a p value of 0.001 I need a chi squared value of <52.620 yet for a p value of 0.995 I need a chi squared value of <10.520. This seems contradictory however given that lower chi squared (sum of squares) corresponds to a 'better' fit. Does this mean I can reject any hypothesis with any chi squared greater than 52.620 with a 0.001 (units?)level of confidence? If this is the case, what then of the 'better' fits? 
 A: You reject the null hypothesis (good fit) for LARGE values
of the computed chi-squared statistic (call it Q). 
Under the null hypothesis, the expected value of Q is the
degrees of freedom. So with DF = 25, you would NEVER reject
for a value of Q less than 25.
If Q = 51.3, the exact P-value is 0.001468724 > 0.001, from
software (still with df = 25). This means that you cannot 
reject at level .001. 
If Q = 53.6, the exact P-value is 0.0007485428 < 0.001, so
you CAN reject at level .001.
If Q = 10.52, then you cannot reject at any reasonable
level of significance, because 10.52 < E(Q) = 25. The
value 10.52 cuts probability .005 from the LOWER tail
of the chi-squared distribution. This might be useful
for some applications of the chi-squared distribution
(for example, finding a confidence bound on the variance of a
normal distribution), but not for a goodness-of-fit (GOF) test.
[Note: In practice, if I got Q as small as 10.52 in a GOF test
with df = 25, I would suspect something is wrong with
either the model or the data. This seems too good a fit to be
true. Sort of analogous to getting reported results 100, 101, 99, 102, 98, 100
on the respective faces 1 through 6 on 600 rolls of a fair die.
Technically, not impossible, but I might suspect someone just
wrote down fake data instead of actually rolling a die. That
would be Q = 0.1 with df = 5.]
There are several statements in your question that
indicate confusion. I have tried to give examples here
that get to point in trying to clear things up. Please
leave a Comment with specific numbers if you have
further questions. I, or someone else, will probably
be able to respond.
Addendum: The figure below shows the density function of CHISQ(df = 25).
The thin black line is at the mean 25, the dotted red lines
are critical values for tests at levels 0.05, 0.01, and 0.001,
respectively (left to right). They are located at
 37.652, 44.314, and 52.620.
The right-hand panel is a
magnification of the curve to the right of Q = 40. It shows more clearly
the tiny area 0.001 to the right of the dotted line at 52.620.
Values of Q > 52.620 lead to rejection of fit at level 0.001.

