Bivariate Category to Numeric Statistics

t-tests & ANOVAs

Measure statistical significance in difference between means.
t-test: 2 categories
ANOVA (analysis of variance): 3 or more categories.
Unlike Pearson r, they do not measure correlation.
Measure if the differences in observations is statistically significant.
ANOVA: indicates difference but not which one.

Example: Test scores

import pandas as pd
import numpy as np

races = ['white', 'African American', 'Latino', 'other']
weights = [0.6, 0.2, 0.15, 0.05]
income = ['0-25k', '26-75k', '75-150k' ]

student = [i for i in range(100)]
score = [i for i in np.random.normal(0.75, 0.1, 100 )]
race = [i for i in np.random.choice(races, size=100, p=weights)]
gender = [i for i in np.random.choice(['male', 'female'], size=100)]
income = [i for i in np.random.choice([1,2,3,4], size=100, p=[0.15, 0.35, 0.35, 0.15])]
class_size = [i for i in np.random.randint(12, 35, size=100)]

df = pd.DataFrame({'student': student, 'score': score, 'race': race, 'gender': gender, 'income': income, 'class_size': class_size})
df.head()

	student	score	race	gender	income	class_size
0	0	0.741935	white	male	3	30
1	1	0.661050	white	male	3	14
2	2	0.612723	white	female	3	32
3	3	0.729796	white	female	2	32
4	4	0.835688	African American	male	2	19

Example: Test scores, cont.

import matplotlib.pyplot as plt
import seaborn as sns

plt.figure(figsize=(8, 6))

men = df.loc[df["gender"] == "male"]
women = df.loc[df["gender"] == "female"]


sns.histplot(data=men, x="score", color="blue", label="Male", kde=True)
sns.histplot(
    data=women,
    x="score",
    color="red",
    label="Female",
    kde=True,
)
plt.axvline(men["score"].mean(), c="b")
plt.axvline(women["score"].mean(), c="r")
plt.title("Test scores by Gender")
plt.legend()
plt.show()

Test Scores by income

ax = sns.displot(data=df, x='score', hue='income', kind='kde', fill=True)
plt.title('Test scores by income')

Text(0.5, 1.0, 'Test scores by income')

t-test output

measure between 2 categories
t-stat: -infinity to infinity
positive result: first group has greater mean
negative result: second group had greater mean
p-value: significance (0.05).

Finding Critical t

degrees of freedom
observations in 1st group (n1)
observations in 2nd group (n2)
df = n1 + n2 -2
alpha = significance value (0.05)
critical t = stats.t.ppf(1-alpha/2, df)

ANOVA output

F-value: 0 to infinity
positive output
p-value: significance (0.05)

Finding the Critical F

Critical F: value over which indicates significance.
Degrees of freedom between (dfb): categories - 1
Degrees of freedom within (dfw): observations - categories.
alpha: (significance standard, for us 0.05)
critical f = stats.f.ppf(1- alpha, dfb, dfw)
outputs the threshold the f-value must surpass.

Let’s get the t-test from our test data

import scipy.stats as stats
men = df.loc[df['gender']=='male', 'score']
women = df.loc[df['gender']=='female', 'score']
result = stats.ttest_ind(men, women)
t,p = result
d = result.df
# result.df = attribute, not part of tuple
# cannot be deconstructed.
print(f't: {t}, p: {p}, df: {d}')

t: 0.032328039796855286, p: 0.9742761987526832, df: 98.0

Example: t-test, cont.

import scipy.stats as stats
# deconstruct the output 
t,p = stats.ttest_ind(men, women)

print(f't: {t}, p: {p}')

t: 0.032328039796855286, p: 0.9742761987526832

Does it surpass critical t

import scipy.stats as stats

d = 98

alpha = 0.05 # set confidence level
criticalt = stats.t.ppf(1 - alpha / 2, d)
print(f'critical-t = {criticalt}, t={t}')

critical-t = 1.9844674545084815, t=0.032328039796855286

ANOVA Example

import scipy.stats as stats

i1 = df.loc[df["income"] == 1, "score"]
i2 = df.loc[df["income"] == 2, "score"]
i3 = df.loc[df["income"] == 3, "score"]
i4 = df.loc[df["income"] == 4, "score"]
f_return = stats.f_oneway(i1, i2, i3, i4)
print(f_return)

F_onewayResult(statistic=np.float64(0.6783918776033178), pvalue=np.float64(0.5673725975511879))

ANOVA Example, faster

import scipy.stats as stats

print("unique values list: ", df["income"].unique())
i1, i2, i3, i4 = [df.loc[df["income"] == i, "score"] for i in df["income"].unique()]
f, p = stats.f_oneway(i1, i2, i3, i4)
print(f"f: {f}, p: {p}")

unique values list:  [3 2 1 4]
f: 0.6783918776033178, p: 0.5673725975511879

ANOVA, critical f

import scipy.stats as stats

dfb = len(df["income"].unique()) - 1
dfw = df["income"].count() - len(df["income"].unique())
alpha = 0.05
f_critical = stats.f.ppf(1 - alpha, dfb, dfw)
print(f_critical)

2.69939259755218