compare frequencies between 6 samples

Question

I'm new to statistics so this might be a really newbie question.. I have a table in R as such:

df <- structure(list(SampleID = structure(1:6, .Label = c("A", "C", 
                "H", "K", "V", "HG"), class = "factor"), freq = c(4L, 13L, 4L, 
                8L, 4L, 1L)), row.names = c(1L, 2L, 3L, 4L, 5L, 
                6L), class = "data.frame")

I want to see if frequencies associated with SampleID's, which represent different conditions, are statistically different from each other. I thought about using the chi-squared test of independence but not sure it is the right test.

What I've done so far is convert this table into a matrix using:

df_mat <- table(df$SampleID, df$freq)

then:

chisq.test(df_mat)

which gave me this result:

Pearson's Chi-squared test
data:  df_mat
X-squared = 18, df = 15, p-value = 0.2627

Can I say that as the p-value 0.26 is greater than the .05 significance level, we do not reject the null hypothesis that the sample type is independent of frequency?

Is there another test that would give a fair comparison of frequencies between the samples?

Answer 1

@whuber is correct. Your tabulation yields:

     1 4 8 13
  A  0 1 0  0
  C  0 0 0  1
  H  0 1 0  0
  K  0 0 1  0
  V  0 1 0  0
  HG 1 0 0  0

This treats the counts (1, 4, 8, 13) as fixed levels of a categorical variable. It seems you want to test if the counts are approximately equal. If we take the sum (26) as the total, out of which, say, 13 became c's, you can test your observed counts against a discrete uniform distribution, which is what would occur if each instance had an equal chance of becoming each sampleID. This is a one-way chi-squared test; it is also called a goodness of fit chi-squared test, as you are testing the fit of your data to the discrete uniform. In R, that would be:

chisq.test(df$freq)

#   Chi-squared test for given probabilities
# 
# data:  df$freq
# X-squared = 15.765, df = 5, p-value = 0.007549

On the other hand, perhaps you had one subject in each condition and counted how many times that person did something. In this case, the conditions are fixed a-priori and the counts are the outcome (the opposite of before). You would need multiple participants to get a measure of variability, but if you assume the Poisson distribution (see my answer here: Help me understand poisson.test?), you could conduct fit a model, since the Poisson has a fixed dispersion. In R, this would be:

m = glm(freq~SampleID, df, family=poisson)
summary(m)
# Call:
# glm(formula = freq ~ SampleID, family = poisson, data = df)
# 
# Deviance Residuals: 
# [1]  0  0  0  0  0  0
# 
# Coefficients:
#               Estimate Std. Error z value Pr(>|z|)   
# (Intercept)  1.386e+00  5.000e-01   2.773  0.00556 **
# SampleIDC    1.179e+00  5.718e-01   2.061  0.03926 * 
# SampleIDH   -1.851e-16  7.071e-01   0.000  1.00000   
# SampleIDK    6.931e-01  6.124e-01   1.132  0.25767   
# SampleIDV   -7.448e-17  7.071e-01   0.000  1.00000   
# SampleIDHG  -1.386e+00  1.118e+00  -1.240  0.21500   
# ---
# Signif. codes:  
# 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
# 
# (Dispersion parameter for poisson family taken to be 1)
# 
#     Null deviance:  1.5278e+01  on 5  degrees of freedom
# Residual deviance: -8.8818e-16  on 0  degrees of freedom
# AIC: 32.151
# 
# Number of Fisher Scoring iterations: 3

From there, you can use the null and residual deviances to conduct a test (see my answer here: Test GLM model using null and model deviances). In R, you would do:

1-pchisq(1.5278e+01, 5)
# [1] 0.009238243

Under either interpretation, your results are significant by conventional criteria.