My understanding of null and alternate hypotheses :

The null hypothesis is a hypothesis whose plausibility is being tested by a test statstic and is denoted by H0. It assumes that observed data follows a standard scientific theory (and the variation/deviation is due to chance).1 Alternative hypothesis is a hypothesis that contradicts null hypothesis. It assumes that the variation in the observed data is real, not due to chance.It is usually the hypothesis that a scientist is trying to establish. (Source: Biostatistics by Lee & Forthofer & Statistics by David Freedman)

This1 holds good for Goodness-of-fit test but not chi-square test for independence as there is no theoretical data to compare the observed data with.

So my question is in a given problem of chi-square test for independence how to determine what the null should be?

  • When what the scientist is trying to establish is mentioned:

    Q : In a study of effectiveness of an antipsychotic drug, patients are treated with the drug and were compared to those receiving a placebo. 698 of 1068 patients were released after taking the placebo while 639 out of 2,127 relapsed after taking the antipsychotic drug. Test the prediction that the antipsychotic drug is significantly more effective in preventing relapse than placebo.

    A: Ho : Edrug = Eplacebo Ha : Edrug > Eplacebo

    (Reasoning: Ha is the hypothesis the scientist is trying to establish and so Ho should be something that opposes the Ha that he has to disprove.)

  • When what the scientist is trying to establish is not mentioned (at least not clearly):

    Q: Among 60 males and 50 females, 25 males and 20 females were with attached ear lobes. Statistically prove whether or not the attached pinna has any relation with sex.

    Not sure what the scientist wants to establish so unable to assume a null.

  • $\begingroup$ What is an attached earlobe? Earlobes are always associated with the pinna right? $\endgroup$ – AliceD Apr 9 '17 at 21:28
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    $\begingroup$ @AliceD It's a commonly used human trait for biology/genetics classes that was thought to follow Mendelian genetics (attached being recessive). It refers to whether the earlobe is attached to the face along its length or whether there is a portion that hangs down below the lowest attachment point of the pinna. Its better known now that it actually isn't a simple Mendelian trait but it still shows up in textbooks as an example. $\endgroup$ – Bryan Krause Apr 9 '17 at 21:39
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    $\begingroup$ @SanjuktaGhosh I don't understand what your point of confusion is. There is a theoretical null when you use chi-square test for independence. Specifically, the null is that the variables are independent. So, in a population that has both males and females and attached and unattached ear lobes you expect the proportion of males with attached ear lobes to be the same as the proportion of females with attached ear lobes. $\endgroup$ – Bryan Krause Apr 9 '17 at 21:43
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    $\begingroup$ This question looks more appropriate for CrossValidated than Biology to me. The questions are framed as biological questions but this is really a question about statistics -- nothing fundamentally "biological" about them. $\endgroup$ – Ben Bolker Apr 9 '17 at 23:00
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    $\begingroup$ @SanjuktaGhosh Yes. Maybe part of the confusion comes from the somewhat poor naming of the tests. In truth, a "test for independence" is really testing for non-independence: that is, you reject the null hypothesis if the samples are not independent. Same thing with tests for homogeneity, normality, etc: in all cases, you are rejecting the null if the data are not homogeneous, not normal, etc. $\endgroup$ – Bryan Krause Apr 10 '17 at 17:51

The Chi-squared test of independence is, as the name suggests, a test of the independence of two outcomes.

Two outcomes are defined as independent if the joint probability of A and B is equal to the product of the probability of A and the probability of B. Or in standard notation, A and B are independent if:

P(A ∩ B) = P(A) * P(B)

from which it follows that:

P(A | B) = P(A)

So in your drug example, there is a probability that a person in the study is given the drug, denoted P(drug), and a probability that a person in the study is released, denoted P(released). The probability of being released is independent of the drug if:

P(drug ∩ released) = P(drug) * P(released)

Release rates can be higher for individuals given the drug, or they can be lower for individuals given the drug, and in either case, release rates would not be independent of drug. So Ha is not

P(released | drug) > P(released)

rather, it is

P(released | drug) ≠ P(released)

In your second example, there is a probability that a person is female, denoted P(female), and a probability that a person has attached earlobes, denoted P(attached). These events are independent if:

P(attached ∩ female) = P(attached) * P(female)

and Ha can be stated as:

P(attached ∩ female) ≠ P(attached) * P(female)

or equivalently:

P(attached | female) ≠ P(attached)
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  • $\begingroup$ Not sure if you missed this. The question in the 1st problem essentially asks to test the hypothesis that the Effectiveness is more. 'Test the prediction that the antipsychotic drug is significantly more effective in preventing relapse than placebo.' $\endgroup$ – Tyto alba Apr 12 '17 at 21:03
  • $\begingroup$ That's the reason I pointed out that a significant result on a chi-squared test of independence is not sufficient to conclude that the drug is more effective than the placebo, only that the effectiveness of the drug and the placebo are not equal. Researchers will often get a statistically-significant 'not equal', and then look at the contrast and interpret it as 'greater than' or 'less than' based on the observed direction, but that is not what the test formally tests. $\endgroup$ – bshane Apr 13 '17 at 0:50
  • $\begingroup$ Could you tell me a bit more about 'Researchers will often get a statistically-significant 'not equal', and then look at the contrast and interpret it as 'greater than' or 'less than' based on the observed direction'? $\endgroup$ – Tyto alba Apr 20 '17 at 14:13
  • $\begingroup$ Some detailed examples are available here: math.la.asu.edu/~elajack/Fall14/STP231/Lecture_ch10.pdf. Essentially, since the test only formally tests 'is response independent of treatment', a researcher who gets a significant result might then look at their data to infer directionality. e.g., a researcher has 100 trials of treatment A, and 100 trials of B. The success-rate is 22/100 for A, and 80/100 for B, so they have a significant result: success-rate depends on treatment. Since 80 > 22, they might further conclude that success is higher for B than for A, with caveats at the link. $\endgroup$ – bshane Apr 21 '17 at 1:55

It might help if you first ask yourself what the independence test is for in plain English, you are testing to see if one categorical variable has an influence on another. In other words, if a person (or whatever)with attribute X (for example, is bald) are they significantly more/less likely to have attribute Y (like being allergic to penicillin) than someone who does not have attribute X.

So in your example the alternative hypothesis would be (in words): the ratio of males with attached ear lobes compared to males without is equal to the ratio of females with attached ear lobes compared to those without.

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