goodness of fit test for grouped data

Question

I have the following grouped data:

Can someone please tell me how to run a goodness of fit test to test if this data is from a normal distribution?

I would like to do this without estimating the parameters from the data.

I could only find PearsonChiSquaredTest, but it is for raw data, not grouped data.

Answer 1

You should still ask this on CrossValidated as I hope you'll get lectures about P-values. Here's one excellent lecture: Frank Harrell on P-values. (In short, one should not rely solely on P-values to make decisions.)

In the meantime here are two approaches to get approximate P-values. First define a function to get the chisquare value:

(* Some data *)
freq = {45, 55, 38, 27, 25, 10};
boundaries = {0, 10, 15, 20, 25, 30, 40};

(* Expand to allow for (nearly) the complete range of a normal distribution *)
freq = Join[{0}, freq, {0}]
(* {0, 45, 55, 38, 27, 25, 10, 0} *)
boundaries = Join[{-$MaxMachineNumber}, boundaries, {$MaxMachineNumber}]
(* {-1.79769*10^308, 0, 10, 15, 20, 25, 30, 40, 1.79769*10^308} *)

chisquare[freq_, boundaries_] := Module[{logL, midPoints, μ0, σ0, sol, d, expected, n},
  logL = Sum[freq[[i]]*
    Log[CDF[NormalDistribution[μ, σ], boundaries[[i + 1]]] - CDF[NormalDistribution[μ, σ], boundaries[[i]]]],
    {i, Length[freq]}];

  (* Get starting values for μ and σ *)
  midPoints = (boundaries[[2 ;; 7]] + boundaries[[3 ;; 8]])/2;
  (μ0 = midPoints.freq[[2 ;; 7]]/Total[freq]);
  (σ0 = Sqrt[((midPoints - μ0)^2).freq[[2 ;; 7]]/Total[freq]]);
  (* Find maximum likelihood estimates of μ and σ *)
  sol = FindMaximum[{logL, σ > 0}, {{μ, μ0}, {σ, σ0}}];

  n = Total[freq];
  d = NormalDistribution[μ, σ] /. sol[[2]];
  expected = Table[n (CDF[d, boundaries[[i + 1]]] - CDF[d, boundaries[[i]]]), {i, 1, 8}];
  {Total[(freq - expected)^2/expected], μ, σ} /. sol[[2]]]

t = chisquare[freq, boundaries]
(* {15.3559, 16.2655, 7.92905} *)

So the observed chisquare value is 15.3559. (The other two values are the respective maximum likelihood estimates of the mean and standard deviation from the grouped data.)

The approximate P-value using the chisquare distribution approximation is given by

1 - CDF[ChiSquareDistribution[Length[freq] - 3], t[[1]]]
(* 0.017662123393318407` *)

as there are 3 restrictions: (1) total count, (2) estimation of mean, and (3) estimation of standard deviation. (Note that I did not combine groups when expected counts are less than 5. One could and should add in that modification.)

A parametric bootstrap approach randomly samples from a normal distribution with mean and standard deviation equal to the maximum likelihood estimates. Then the chisquare statistic is calculated for each sample. The estimated P-value is proportion of bootstrap chisquare values greater than or equal to the observed chisquare value.

(* Now perform a parametric bootstrap to estimate P-value *)
d = NormalDistribution[t[[2]], t[[3]]];
n = Total[freq];
nBoot = 1000;
pValue = 0;
SeedRandom[12345];
Do[x = RandomVariate[d, n];
 h = HistogramList[x, {boundaries}];
 c = chisquare[h[[2]], h[[1]]];
 If[c[[1]] >= t[[1]], pValue = pValue + 1],
 {i, nBoot}]
(pValue = pValue/nBoot) // N
(* 0.059` *)