GW Blok 6 Seminar 4.2 Chi-square test for categorical variables

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What is the chi-square test of independence?

  • To test whether two categorical variables are statistically associated (i.e., not independent)

  • This test statistic can safely be used, if all expected counts are larger than 5 (see bottom table under sub a.) Then use asymptomatic sig. for p-value

    • Otherwise use the exact sig. for p-value

  • Degrees of freedom: (number of rows - 1) x (numer of columns - 1)

  • If p-value < 0.05 → Reject H₀

    • There is a significant association

  • If p-value ≥ 0.05 → Do not reject H₀

    • No significant association

  • A large chi-square → more difference between observed & expected → more evidence against H₀

<ul><li><p>To test whether <strong>two categorical variables</strong> are <strong>statistically associated</strong> (i.e., not independent)</p></li><li><p>This test statistic can safely be used, if all expected counts are larger than 5 (see bottom table under sub a.) Then use asymptomatic sig. for p-value</p><ul><li><p>Otherwise use the exact sig. for p-value</p></li></ul></li><li><p>Degrees of freedom: (number of rows - 1) x (numer of columns - 1)</p></li><li><p>If <strong>p-value &lt; 0.05</strong> → Reject H₀</p><ul><li><p> There <strong>is a significant association</strong></p></li></ul></li><li><p>If <strong>p-value ≥ 0.05</strong> → Do not reject H₀</p><ul><li><p>No significant association</p></li></ul></li><li><p>A large chi-square → more difference between observed &amp; expected → more evidence <strong>against</strong> H₀</p></li></ul><p></p>
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What are the different ways to write hypotheses? (probability function, RR, OR)

In words:

  • H₀: No relationship between the two variables → they are independent

  • Hₐ: There is a relationship → they are not independent

Example – Social Distancing & Study Program: a) Probability formulation:

  • H₀: π(social distancing = yes | EPH1026) = π(social distancing = yes | GZW1026)

  • Hₐ: π(social distancing = yes | EPH1026) ≠ π(social distancing = yes | GZW1026)

b) Relative Risk (RR):

  • H₀: RR = 1

  • Hₐ: RR ≠ 1

c) Odds Ratio (OR):

  • H₀: OR = 1

  • Hₐ: OR ≠ 1

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What are the assumption of the chi-square test?

  • Groups are independent

  • Observations are independent

  • Every person occurs only once in one cell of the cross table. Hence no repeated measurements for 1 person.

    • This is an assumption

  • Expected frequencies (E) ≥ 5

    • Not an assumption

If not satisfied: better to employ an exact test (Fisher’s)

We can still do the chi square test if there are more than 2 levels (ex. not vaccinated, a little, a lot)

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How can you quantify the strength of an association?

  • To quantify the strength of association, use RR or OR

    • You can’t calculate the RR with a case control study design

    • You can always calculate the OR

  • Remember in the sample:

    • The probability (“risk”) of supporting social distancing after vaccination for EPH1026 is 2.3 times the probability (“risk”) for GZW1026

    • The odds of supporting social distancing after vaccination in EPH1026 is 3.3 times the odds in GZW1026

  • At the population level, there is evidence that RR≠1 (i.e., OR ≠1). Why?

    • We are rejecting H0 → we assume a relationship