Getting p-value=1 on a Goodness to fit Chi squared test

1
votes

I'm trying to do a goodness-to-fit test against a Poisson on a series of observations using R. I'm counting how many people did a certain thing per minute, over 57 minutes. I never got any observations greater than 13, and i got the following data: (for the cases 0 to 13+ people):

observed = c(3/57, 4/57, 9/57, 7/57, 9/57, 8/57, 2/57, 3/57, 7/57, 2/57, 1/57, 0, 1/57, 1/57, 0)

meaning that 3 times i observed 0 people, 4 times 1 people, 9 times 2 people and so on (the last 0 means i never saw 14 or more people).

mn = 4.578947 
cases = c(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13)
estimated = c()
for (i in cases)(estimated <- c(estimated, dpois(i, lambda = mn)))
estimated <- c(estimated, (1-ppois(13, lambda=mn)))

where mn is the mean obtained from the data. Finally, i run

 chisq.test(observed, p=estimated)

and i get:

 Chi-squared test for given probabilities

data:  observed
X-squared = 1.0182, df = 14, p-value = 1

Warning message:
In chisq.test(observed, p = estimated) :
  Chi-squared approximation may be incorrect

I'm not well-versed in this area (neither statistics nor programming on R), but i have the idea that i'm not supposed to get a p-value of exactly 1.0. What am i doing wrong? (By the way: My code is most likely not optimal for what i'm trying to do, but i barely use R and it's not the focus of my work right now.)

2
rstatisticschi-squaredgoodness-of-fit
In addition to use counts data for observed frequencies, you need to have expected frequencies >= 5 for each bin/ category of occurrence. explained in my answer below on how to achieve that. - Mankind_008

2 Answers

2
votes

Your observed values should be counts, not proportions:

> chisq.test(observed*57, p=estimated)

    Chi-squared test for given probabilities

data:  observed * 57
X-squared = 58.036, df = 14, p-value = 2.585e-07

Per the R help file for chisq.test:

If x is a matrix with one row or column, or if x is a vector and y is not given, then a goodness-of-fit test is performed (x is treated as a one-dimensional contingency table). The entries of x must be non-negative integers.

(Emphasis mine)

You can test this with some of the example code in the manual

How it should be done:

> x <- c(89,37,30,28,2)
> p <- c(0.40,0.20,0.20,0.19,0.01)
> chisq.test(x, p = p)

    Chi-squared test for given probabilities

data:  x
X-squared = 5.7947, df = 4, p-value = 0.215

Warning message:
In chisq.test(x, p = p) : Chi-squared approximation may be incorrect

And making the same mistake as you have:

> chisq.test(x/sum(x), p = p)

    Chi-squared test for given probabilities

data:  x/186
X-squared = 0.031154, df = 4, p-value = 0.9999

Warning message:
In chisq.test(x/186, p = p) : Chi-squared approximation may be incorrect
0
votes

Firstly, to conduct a goodness of fit test, observed frequencies and bin probabilities are required.

 observed = c(3, 4, 9, 7, 9, 8, 2, 3, 7, 2, 1, 0, 1, 1, 0)       # keep counts

Probabilities are correct:

 mn = 4.578947 
 prob = c()
 for (i in cases)     (prob <- c(prob, dpois(i, lambda = mn)))
 prob <- c(prob, (1-ppois(13, lambda=mn)))           # prob for 13 and plus category

Most importantly, Expected frequencies in a bin/ category should be at least 5. Chisq-test is not valid for small samples. This is why you get a warning (see expected frequencies for category 1,2 and 8-15) :

poisson_df <- data.frame(observed, prob)
poisson_df$expected = sum(poisson_df$observed)*poisson_df$prob

poisson_df

#   observed   prob          expected
#1         3   0.0102657004  0.58514492
#2         4   0.0470060980  2.67934759
#3         9   0.1076192157  6.13429530
#4         7   0.1642608950  9.36287101
#5         9   0.1880354831 10.71802253
#6         8   0.1722009022  9.81545143
#7         2   0.1314164674  7.49073864
#8         3   0.0859641485  4.89995646
#9         7   0.0492031600  2.80458012
#10        2   0.0250331846  1.42689152
#11        1   0.0114625626  0.65336607
#12        0   0.0047714970  0.27197533
#13        1   0.0018207026  0.10378005
#14        1   0.0006413001  0.03655410
#15        0   0.0002986829  0.01702492

chisq.test(x = poisson_df$observed, p= poisson_df$prob)

# Chi-squared test for given probabilities

# data:  observed
# X-squared = 58.036, df = 14, p-value = 2.585e-07

Warning message:
In chisq.test(x = poisson_df$observed, p= poisson_df$prob) :
Chi-squared approximation may be incorrect

Therefore, you need to create bins appropriately. It should be noted that Chisq-test is sensitive to binning, one way to bin is as below:

cat_eq_3_less <- apply(poisson_df[1:3,], 2 , sum)        # sum of 1 to 3 categories
cat_eq_8_plus <- apply(poisson_df[8:15,], 2 , sum)       # sum 8 to 15 categories

corrected_df <- rbind(cat_eq_3_less, poisson_df[4:7,], cat_eq_8_plus)

 corrected_df
 #   observed     prob       expected
 #        16      0.1648910  9.398788
 #         7      0.1642609  9.362871
 #         9      0.1880355 10.718023
 #         8      0.1722009  9.815451
 #         2      0.1314165  7.490739
 #        15      0.1791952 10.214129

chisq.test(x = corrected_df$observed, p = corrected_df$prob)

Chi-squared test for given probabilities

data:  corrected_df$observed
X-squared = 12.111, df = 5, p-value = 0.0333