Chi-Square Independence

197. Chi-Square Independence

Compares two observed categorical distributions.

Χ^{2} = \sum_{𝑖 = 1}^{𝑘} \frac{{(𝑜_{𝑖 𝑗} - 𝑒_{𝑖 𝑗})}^{2}}{𝑒_{𝑖 𝑗}}

$𝑜_{𝑖 𝑗}$ : observed frequency in cell $(𝑖, 𝑗)$
$𝑒_{𝑖 𝑗}$ : expected frequency in category $(𝑖, 𝑗)$ calculated as:

𝐸_{𝑖 𝑗} = \frac{𝑅_{𝑖} \times 𝐶_{𝑗}}{𝑁}

$𝑅_{𝑖}$ : Row total for row $𝑖$
$𝐶_{𝑗}$ : Column total for column $𝑗$
$𝑁$ : Total number of observations

df = (rows - 1) (cols - 1)

Example

You own a café, and you think you can make more money by getting people to buy larger drinks in the morning. You want to see if the time of day and coffee size are related, so you collect a simple random sample of last month’s orders. The observed data is below. At $𝛼 = 0.05$ test if they are related.

Observed	Small	Medium	Large	Total
Morning	7	14	29	50
Afternoon	7	7	15	29
Evening	10	7	4	21
Total	24	28	48	100

expected = \frac{row total \times col total}{table total}

Expected	Small	Medium	Large	Total
Morning	$\frac{50 \cdot 24}{100} = 12$	$\frac{50 \cdot 28}{100} = 14$	$\frac{50 \cdot 48}{100} = 24$	50
Afternoon	$\frac{29 \cdot 24}{100} = 6.96$	$\frac{29 \cdot 28}{100} = 8.12$	$\frac{29 \cdot 48}{100} = 13.92$	29
Evening	$\frac{21 \cdot 24}{100} = 5.04$	$\frac{21 \cdot 28}{100} = 5.88$	$\frac{21 \cdot 48}{100} = 10.08$	21
Total	24	28	48	100

\begin{aligned} 𝐻_{0} & : the two variables are independent \\ 𝐻_{𝑎} & : the 2 variables are related \end{aligned}

\begin{aligned} 𝜒^{2} & = & \frac{{(7 - 12)}^{2}}{12} + \frac{{(14 - 14)}^{2}}{14} + \frac{{(29 - 24)}^{2}}{24} + \\ \frac{{(7 - 6.96)}^{2}}{6.96} + \frac{{(7 - 8.12)}^{2}}{8.12} + \frac{{(15 - 13.92)}^{2}}{13.92} + \\ \frac{{(10 - 5.04)}^{2}}{5.04} + \frac{{(7 - 5.88)}^{2}}{5.88} + \frac{{(4 - 10.08)}^{2}}{10.08} \\ = & 12.13 \end{aligned}

p-value < alpha

0.0164 < 0.05

Critical value < Chi^2 statistic

9.49 < 12.13

Example

Genre Preference by Age Group

	Action	Comedy	Drama
Under 30	60	50	30	140
Above 30	20	40	60	120
	80	90	90	260

	Action	Comedy	Drama
Under 30	$\frac{140 \cdot 80}{260}$	$\frac{140 \cdot 90}{260}$	$\frac{140 \cdot 90}{260}$	140
Above 30	$\frac{120 \cdot 80}{260}$	$\frac{120 \cdot 90}{260}$	$\frac{120 \cdot 90}{260}$	120
	80	90	90	260

	Action	Comedy	Drama
Under 30	$43.08$	$48.46$	$48.46$	140
Above 30	$36.92$	$41.54$	$41.54$	120
	80	90	90	260

	Action	Comedy	Drama
Under 30	$\frac{{(60 - 43.08)}^{2}}{43.08}$	$\frac{{(50 - 48.46)}^{2}}{48.46}$	$\frac{{(30 - 48.46)}^{2}}{48.46}$	140
Above 30	$\frac{{(20 - 36.92)}^{2}}{36.92}$	$\frac{{(40 - 41.54)}^{2}}{41.54}$	$\frac{{(60 - 41.54)}^{2}}{41.54}$	120
	80	90	90	260

	Action	Comedy	Drama	$\sum$
Under 30	$6.65$	$0.05$	$7.03$	$13.73$
Above 30	$7.76$	$0.06$	$8.21$	$16.02$
$\sum$	$14.40$	$0.11$	$15.24$	$𝟐𝟗.𝟕𝟓$

{Chi}^{2} = 6.65 + 0.05 + 7.03 + 7.76 + 0.06 + 8.21 = 29.75

Since $Χ^{2} = 29.75 > Χ_{0.05, 2}^{2} = 5.991$ , reject $𝐻_{0}$ : genre preference depends on age group, driven almost entirely by action and drama ( $14.40$ and $15.24$ ) rather than comedy ( $0.11$ ).