First of all, I’d like to ask how you would like to estimate the chance of having nukes in a country? What this 80% implies here? But, before getting to the question, I’d like to discuss computing the chance of e coli infection, first.

From the frequentists perspective, computing the chance of e coli infection is investigating sample of lettuce and counts species that are infected: n is the number of infected species and N is the total sample size. 1% means one among 100. Such percentage reports and their uncertainties are very familiar scene during any election periods for everyone. From Bayesian perspective, Pr(p|D) ~ L(D|p) pi(p), properly choosing likelihoods and priors, one can estimate the chance of e coli infection and uncertainty. Understanding of sample species and a prior knowledge helps to determine likelihoods and priors.

How about the chance that country A has nukes? Do we have replicates of country A so that a committee investigate each country and count ones with nukes to compute the chance? We cannot do that. Traditional frequentist approach, based on counting, does not work here to compute the chance. Either using fiducial likelihood approach or Bayesian approach, i.e. carefully choosing a likelihood function adequately (priors are only for Bayesian) allows one to compuate such probability of interest. In other words, those computed chances highly depend on the choice of model and are very subjective.

So, here’s my concern. It seems like that astronomers want to know the chance of their spectral data being described by a model (A*B+C)*D (each letter stands for one of models such as listed in Sherpa Models). This is more like computing the chance of having nukes in country A, not counting frequencies of the event occurrence. On the other hand, p-value from goodness of fit tests, LRTs, or F-tests is a number from the traditional frequentists’ counting approach. In other words, p-value accounts for, under the null hypothesis (the (A*B+C)*D model is the right choice so that residuals are Gaussian), how many times one will observe the event (say, reduced chi^2 >1.2) if the experiments are done N times. The problem is that we only have one time experiment and that one spectrum to verify the (A*B+C)*D is true. Goodness of fit or LRT only tells the goodness or the badness of the model, not the statistically and objectively quantified chance.

In order to know the chance of the model (A*B+C)*D, like A has nuke with p%, one should not rely on p-values. If you have multiple models, one could compute pairwise relative chances i.e. odds ratios, or Bayes factors. However this does not provide the uncertainty of the chance (astronomers have the tendency of reporting uncertainties of any point estimates even if the procedure is statistically meaningless and that quantified uncertainty is not statistical uncertainty, as in using delta chi^2=1 to report 68% confidence intervals). There are various model selection criteria that cater various conditions embedded in data to make a right model choice among other candidate models. In addition, post-inference for astronomical models is yet a very difficult problem.

In order to report the righteous chance of (A*B+C)*D requires more elaborated statistical modeling, always brings some fierce discussions between frequentists and Bayesian because of priors and likelihoods. Although it can be very boring process, I want astronomers to leave the problem to statisticians instead of using inappropriate test statistics and making creative interpretation of statistics.

Please, keep this question in your mind when you report probability: what kind of chance are you computing? The chance of e coli infection? Or the chance that A has nukes? Make sure to understand that p-values from data analysis packages does not tell you that the chance the model (A*B+C)*D is (one minus p-value)%. You don’t want to report one minus p-value from a chi-square test statistic as the chance that A has nukes.

Full text search in ADS with “null hypothesis probability” yield 77 related articles (link removed. Search results are floating urls, probably?). Many of them contained the phrase “null hypothesis probability” as it is. The rest were in the context of “given the null hypothesis, the probability of …” I’m not sure this ADS search result includes null hypothesis probability written in tables and captions. It’s possible more than 77 could exist. The majority of articles with the “null hypothesis probability” are just reporting numbers from screen outputs from the chosen data analysis system. Discussions and interpretations of these numbers are more focused toward reduced χ² close to ONE, astronomers’ most favored model selection criterion. Sometimes, I got confused with the goal of their fitting analysis because the driven force is that “make the reduced chi-square closed to one and make residuals look good“. Instead of being used for statistical inferences and measures, a statistic works as an objective function. Numerically (chi-square) or pictorially (residuals) is overshadowed the fundamentals that you observed relatively low number of photons under Poisson distribution and those photons are convolved with complicated instruments. It is possible to underestimated statistically, the reduced chi-sq is off from the unity but based on robust statistics, one still can say the model is a good fit.

Instead of talking about the business of the chi-square method, one thing I wanted to point out from this “null hypothesis probability” investigation is that there was a big presenting style and field distinction between papers of the null hypothesis probability (spectral model fitting) and of given the null hypothesis, the probability of (cosmology). Beyond this casual and personal finding about the style difference, the following quotes despaired me because I couldn’t find answers from statistics.

• MNRAS, v.340, pp.1261-1268 (2003): The temperature and distribution of gas in CL 0016+16 measured with XMM-Newton (Worrall and Birkinshaw)
  

  With reduced chi square of 1.09 (chi-sq=859 for 786 d.o.f) the null hypothesis probability is 4 percent but this is likely to result from the high statistical precision of the data coupled with small remaining systematic calibration uncertainties

  I couldn’t understand why p-value=0.04 is associated with high statistical precision of the data coupled with small remaining systematic calibration uncertainties. Is it a polite way to say the chi-square method is wrong due to systematic uncertainty? Or does this mean the stat uncertainty is underestimated due the the correlation with sys uncertainty? Or other than p-value, does the null hypothesis probability has some philosophical meanings? Or … I may go on with strange questions due to the statistical ambiguity of the statement. I’d appreciate any explanation how the p-value (the null hypothesis probability) is associated with the subsequent interpretation.

  Another miscellaneous question is that If the number (the null hypothesis probability) from software packages is unfavorable or uninterpretable, can we attribute such ambiguity to systematical error?

• MNRAS, v. 345(2),pp.423-428 (2003): Iron K features in the hard X-ray XMM-Newton spectrum of NGC 4151 (Schurch, Warwick, Griffiths, and Sembay)
  

  The result of these modifications was a significantly improved fit (chi-sq=4859 for 4754 d.o.f). The model fit to the data is shown in Fig. 3 and the best-fitting parameter values for this revised model are listed as Model 2 in Table 1. The null hypothesis probability of this latter model (0.14) indicates that this is a reasonable representation of the spectral data to within the limits of the instrument calibration.

  What is the rule of thumb interpretation of p-values or this null hypothesis probability in astronomy? How one knows that it is reasonable as authors mentioned? How one knows the limits of the instrument calibration and compares quantitatively? How about the degrees of freedom? Some thousands! So large. Even with a million photons, according to the guideline for the number of bins^[1] I doubt that using chi-square goodness of fit for data with such large degree of freedom makes the test too conservative. Also, there should be distinction between the chi square minimization tactic and the chi square goodness of fit test. Using same data for both procedures will introduce bias.

• MNRAS, v. 354, pp.10-24 (2004): Comparing the temperatures of galaxy clusters from hdrodynamical N-body simulations to Chandra and XMM-Newton observations (Mazzotta, Rasia, Moscardini, and Tormen)
  

  In particular, solid and dashed histograms refer to the fits for which the null hypothesis has a probiliy >5 percent (statistically acceptable fit) or <5 percent (statistically unacceptable fit), respectively. We also notice that the reduced chi square is always very close to unity, except in a few cases where the lower temperature components is at T~2keV, …

  The last statement obscures the interpretation even more to the statement related to what “statistically (un)acceptable fit” really means. The notion of how good a model fits to data and how to test such hypothesis from the statistics standpoint seems different from that of astronomy.

• MNRAS, v.346(4),pp.1231-1241: X-ray and ultraviolet observations of the dwarf nova VW Hyi in quiescence (Pandel, Córdova, and Howell)
  

  As can be seen in the null hypothesis probabilities, the cemekl model is in very good agreement with the data.

  The computed null hypothesis probabilities from the table 1 are 8.4, 25.7, 42.2, 1.6, 0.7*, and 13.1 percents (* is the result of MKCFLOW model, the rest are CEMEKL model). Probably, the criterion to declare a good fit is a p-value below 0.01 so that CEMEKL model cannot be rejected but MKCFLOW model can be rejected. Only one MKCFLOW which by accident resulted in a small p-value to say that MKCFLOW is not in agreement but the other choice, CEMEKL model is a good model. Too simplified model selection/assessment procedure. I wonder why CEMEKL was tried with various settings but MKCFLOW was only once. I guess there’s is an astrophysical reason of executing such model comparison study but statistically speaking, it looks like comparing measurement of 5 different kinds of oranges and one apple measured by a same ruler (the null hypothesis probability from the chi-square fitting). From the experimental design viewpoint, this is not well established study.

• MNRAS, 349, 1267 (2004): Predictions on the high-frequency polarization properties of extragalactic radio sources and implications for polarization measurements of the cosmic microwave background (Tucci et al.)
  

  The correlation is less clear in the samples at higher frequencies (r~ 0.2 and a null-hypothesis probability of >10^{-2}). However, these results are probably affected by the variability of sources, because we are comparing data taken at different epochs. A strong correlation (r>0.5 and a null-hypothesis probability of <10^{-4}) between 5 and 43 GHz is found for the VLA calibrators, the polarization of which is measured simultaneously at all frequencies.

  I wonder what test statistic has been used to compute those p-values. I wonder if they truly meant p-value>0.01. At this level, most tools offer more precise number so as to make a suitable statement. The p-value (or the “null hypothesis probability”) is for testing whether r=0 or not. Even r is small, 0.2, still one can reject the null hypothesis if the threshold is 0.05. Therefore, >10^{-2} only add ambiguity. I think point estimates are enough to report the existence of weak and rather strong correlations. Otherwise, reporting both p-values and powers seems more appropriate.

• A&A, 342, 502 (1999): X-ray spectroscopy of the active dM stars: AD Leo and EV Lac
  (S. Sciortino, A. Maggio, F. Favata and S. Orlando)

  This fit yields a value of chi square of 185.1 with 145 υ corresponding to a null-hypothesis probability of 1.4% to give an adequate description of the AD Leo coronal spectrum. In other words the adopted model does not give an acceptable description of available data. The analysis of the uncertainties of the best-fit parameters yields the 90% confidence intervals summarized in Table 5, together with the best-fit parameters. The confidence intervals show that we can only set a lower-limit to the value of the high-temperature. In order to obtain an acceptable fit we have added a third thermal MEKAL component and have repeated the fit leaving the metallicity free to vary. The resulting best-fit model is shown in Fig. 7. The fit formally converges with a value of chi square of 163.0 for 145 υ corresponding to a probability level of ~ 9.0%, but with the hotter component having a “best-fit” value of temperature extremely high (and unrealistic) and essentially unconstrained, as it is shown by the chi square contours in Fig. 8. In summary, the available data constrain the value of metallicity to be lower than solar, and they require the presence of a hot component whose temperature can only be stated to be higher than log (T) = 8.13. Available data do not allow us to discriminate between the (assumed) thermal and a non-thermal nature of this hot component.
  …The fit yields a value of [FORMULA] of 95.2 (for 78 degree of freedom) that corresponds to a null hypothesis probability of 2.9%, i.e. a marginally acceptable fit. The limited statistic of the available spectra does not allow us to attempt a fit with a more complex model.

  After adding MEKAL, why the degree of freedom remains same? Also, what do they mean by the limited statistic of the available spectra?

• MNRAS348, 529 (2004):Powerful, obscured active galactic nuclei among X-ray hard, optically dim serendipitous Chandra sources (Gandhi, Crawford, Fabian, Johnstone)
  

  …, but a low f-test probability for this suggests that we cannot constrain the width with the current data.
  While the rest frame equivalent width of the line is close to 1keV, its significance is marginal (f-test gives a null hypothesis probability of 0.1).

  Without a contingency table, nor comparing models, I was not sure how they executed the F-test. I could not find two degrees of freedom for the F-test. From the XSPEC’s account for the F-test (http://heasarc.gsfc.nasa.gov/docs/xanadu/xspec/manual/XSftest.html), we see two degrees of freedom, without them, no probability can be computed. Their usage of the F-test seems unconventional. The conventional application of the F-test is for comparing effects of multiple treatments (different levels of drug dosage including placebo); otherwise, it’s just a chi square goodness of fit test or t-test.

• Another occasion I came across is interpreting the null hypothesis probability of 0.99 as an indicator of a good fit; well, it’s overfitting. Not only too small null hypothesis probability but also close to one null hypothesis probability should raise a flag for cautions and warnings because the later indicating you are overdoing (too many free parameters for example).

There are some residuals of ambiguity after deducing the definition of the null hypothesis probability by playing with XSPEC and finding cases how this null hypothesis probability is used in literature. Authors sometimes added creative comments in order to interpret the null hypothesis probability from their data analysis, which I cannot understand without statistical imagination. Most can be overlooked, perhaps. Or instead, they are rather to be addressed to astronomers with statistical knowledge to resolve my confusion by the null hypothesis probability. I expect comments on how to view these quotes with statistical rigor from astronomers. The listed are personal. There are some more I really didn’t understand the points but many were straightforward in using the null hypothesis probabilities as p-values in statistical inference under the simple null hypothesis. I just listed some to display my first impression on these quotes most of which I couldn’t draw statistical caricatures out of them. Eventually, I hope some astronomers straighten the meaning and the usage of the null hypothesis probability without overruling basics in statistics.

I only want to add a caution when using the reduced chi-square as a model selection criteria. An indicator of a good-fit from a reduced chi^2 close to unity is only true when grouped data are independent so that the formula of degrees of freedom, roughly, the number of groups minus the number of free parameters, is valid. Personally I doubt this rule applied in spectral fitting that one cannot expect independence between two neighboring bins. In other words, given a source model and given total counts, two neighboring observations (counts in two groups) are correlated. The grouping rules like >25 or S/N>3 do not guarantee the independent assumption for the chi-square goodness of fit test although it may sufficient for Gaussian approximation. Statisticians devised various penalty terms and regularization methods for model selection that suits data types. One way to look is computing proper degrees of freedom, called effective degrees of freedom instead of n-p, to reflect the correlation across groups because of the chosen source model and calibration information. With a large number of counts or large number of groups, unless properly penalized, it is likely that the chi-square fit is hard to reject the null hypothesis than a statistic with smaller degrees of freedom because of the curse of dimensionality.

1. Mann and Wald (1942), “On the Choice of the Number of Class Intervals in the Application of the Chi-square Test” Annals of Math. Stat. vol. 13, pp.306-7.

Systematic Errors by J. Heinrich and L.Lyons
in Annu. Rev. Nucl. Part. Sci. (2007) Vol. 57 pp.145-169 [http://adsabs.harvard.edu/abs/2007ARNPS..57..145H]

The characterization of two error types, systematic and statistical error is illustrated with an simple physics experiment, the pendulum. They described two distinct sources of systematic errors.

…the reliable assessment of systematics requires much more thought and work than for the corresponding statistical error.
Some errors are clearly statistical (e.g. those associated with the reading errors on T and l), and others are clearly systematic (e.g., the correction of the measured g to its sea level value). Others could be regarded as either statistical or systematic (e.g., the uncertainty in the recalibration of the ruler). Our attitude is that the type assigned to a particular error is not crucial. What is important is that possible correlations with other measurements are clearly understood.

Section 2 contains a very nice review in english, not in mathematical symbols, about the basics of Bayesian and frequentist statistics for inference in particle physics with practical accounts. Comparison of Bayes and Frequentist approaches is provided. (I was happy to see that χ² is said to not belong to frequentist methods. It is just a popular method in references about data analysis in astronomy, not in modern statistics. If someone insists, statisticians could study the χ² statistic under some assumptions and conditions that suit properties of astronomical data, investigate the efficiency and completeness of grouped Poission counts for Gaussian approximation within the χ² minimization process, check degrees of information loss, and so forth)

To a Bayesian, probability is interpreted as the degree of belief in a statement. …
In contast, frequentists define probability via a repeated series of almost identical trials;…

Section 3 clarifies the notion of p-values as such:

It is vital to remember that a p-value is not the probability that the relevant hypothesis is true. Thus, statements such as “our data show that the probability that the standard model is true is below 1%” are incorrect interpretations of p-values.

This reminds me of the null hypothesis probability that I often encounter in astronomical literature or discussions to report the X-ray spectral fitting results. I believe astronomers using the null hypothesis probability are confused between Bayesian and frequentist concepts. The computation is based on the frequentist idea, p-value but the interpretation is given via Bayesian. A separate posting on the null hypothesis probability will come shortly.

Section 4 describes both Bayesian and frequentist ways to include systematics. Through its parameterization (for Gaussian, parameterization is achieved with additive error terms, or none zero elements in full covariance matrix), systematic uncertainty is treated as nuisance parameters in the likelihood for both Bayesian and frequentist alike although the term “nuisance” appears in frequentist’s likelihood principles. Obtaining the posterior distribution of a parameter(s) of interest requires marginalization over uninteresting parameters which are seen as nuisance parameters in frequentist methods.

The miscellaneous section (Sec. 6) is the most useful part for understanding the nature and strategies for handling systematic errors. Instead of copying the whole section, here are two interesting quotes:

When the model under which the p-value is calculated has nuisance parameters (i.e. systematic uncertainties) the proper computation of the p-value is more complicated.

The contribution form a possible systematic can be estimated by seeing the change in the answer a when the nuisance parameter is varied by its uncertainty.

As warned, it is not recommended to combine calibrated systematic error and estimated statistical error in quadrature, since we cannot assume those errors are uncorrelated all the time. Except the disputes about setting a prior distribution, Bayesian strategy works better since the posterior distribution is the distribution of the parameter of interest, directly from which one gets the uncertainty in the parameter. Remember, in Bayesian statistics, parameters are random whereas in frequentist statistics, observations are random. The χ² method only approximates uncertainty as Gaussian (equivalent to the posterior with a gaussian likelihood centered at the best fit and with a flat prior) with respect to the best fit and combines different uncertainties in quadrature. Neither of strategies is superior almost always than the other in a general term of performing statistical inference; however, case-specifically, we can say that one functions better than the other. The issue is how to define a model (distribution, distribution family, or class of functionals) prior to deploying various methodologies and therefore, understanding systematic errors in terms of model, or parametrization, or estimating equation, or robustness became important. Unfortunately, systematic descriptions about systematic errors from the statistical inference perspective are not present in astronomical publications. Strategies of handling systematic errors with statistical care are really hard to come by.

Still I think that their inclusion of systematic errors is limited to parametric methods, in other words, without parametrization of systematic errors, one cannot assess/quantify systematic errors properly. So, what if such parametrization of systematics is not available? I thought that some general semi-parametric methodology possibly assists developing methods of incorporating systematic errors in spectral model fitting. Our group has developed a simple semi-parametric way to incorporate systematic errors in X-ray spectral fitting. If you like to know how it works, please check out my poster in pdf. It may be viewed too conservative as if projection since instead of parameterizing systemtatics, the posterior was empirically marginalized over the systematics, the hypothetical space formed by simulated sample of calibration products.

I believe publications about handling systematic errors will enjoy prosperity in astronomy and statistics as long as complex instruments collect data. Beyond combining in quadrature or Gaussian approximation, systematic errors can be incorporated in a more sophisticated fashion, parametrically or nonparametrically. Particularly for the latter, statisticians knowledge and contributions are in great demand.