SMILE — Hypothesis Testing

The smile.stat.hypothesis package provides parametric and non-parametric statistical hypothesis tests. The smile.stat.Hypothesis interface acts as a concise facade that groups all tests under a single import.

Overview
The Hypothesis Facade
t-Tests
F-Tests and ANOVA
Kolmogorov-Smirnov Test
Correlation Tests
Chi-Square Tests
Interpreting Results
Quick Reference

1. Overview

Every test returns a result record that bundles the test statistic, degrees of freedom, and two-sided p-value:

Test	Result type	Key fields
t-test	`TTest`	`t()`, `df()`, `pvalue()`
F-test / ANOVA	`FTest`	`f()`, `df1()`, `df2()`, `pvalue()`
KS test	`KSTest`	`d()`, `pvalue()`
Correlation	`CorTest`	`cor()`, `t()`, `df()`, `pvalue()`
Chi-square	`ChiSqTest`	`chisq()`, `df()`, `pvalue()`, `CramerV()`

All records implement toString() that prints the full test summary.

Significance Codes

The helper Hypothesis.significance(double pvalue) returns the conventional significance annotation:

Range	Code
p < 0.001	`***`
p < 0.01	`**`
p < 0.05	`*`
p < 0.1	`.`
p ≥ 0.1	(blank)

java

double p = 0.003;
System.out.println(Hypothesis.significance(p)); // "**"

2. The `Hypothesis` Facade

smile.stat.Hypothesis is a marker interface with nested static-method interfaces. Import it once and access all tests through a uniform API.

java

import smile.stat.Hypothesis;
import smile.stat.hypothesis.*;

// t-test
TTest t = Hypothesis.t.test(x, 0.0);          // one-sample
TTest t = Hypothesis.t.test(x, y);            // two-sample (Welch)
TTest t = Hypothesis.t.test(x, y, "equal.var");
TTest t = Hypothesis.t.test(x, y, "paired");

// F-test / ANOVA
FTest f = Hypothesis.F.test(x, y);            // variance test
FTest f = Hypothesis.F.test(labels, values);   // one-way ANOVA

// KS test
KSTest ks = Hypothesis.KS.test(x, dist);      // one-sample
KSTest ks = Hypothesis.KS.test(x, y);         // two-sample

// Correlation tests
CorTest c = Hypothesis.cor.test(x, y);                // Pearson
CorTest c = Hypothesis.cor.test(x, y, "spearman");
CorTest c = Hypothesis.cor.test(x, y, "kendall");

// Chi-square tests
ChiSqTest cs = Hypothesis.chisq.test(bins, prob);        // one-sample
ChiSqTest cs = Hypothesis.chisq.test(bins1, bins2);      // two-sample
ChiSqTest cs = Hypothesis.chisq.test(contingencyTable);  // independence

3. t-Tests

Student's t-tests are used to compare means when the population is assumed normally distributed. All variants are in TTest.

One-Sample t-Test

Null hypothesis H₀: the population mean equals a specified value μ₀.

java

double[] x = {2.1, 2.4, 1.9, 2.3, 2.0, 2.2, 1.8, 2.1, 2.0, 2.3};

TTest result = TTest.test(x, 2.0); // H0: mean = 2.0
// or via facade:
TTest result = Hypothesis.t.test(x, 2.0);

System.out.println(result);
// → One Sample t-test(t = 1.1547, df = 9.000, p-value = 0.278 )

System.out.printf("t=%.4f, df=%.0f, p=%.4f%n",
    result.t(), result.df(), result.pvalue());

if (result.pvalue() < 0.05) {
    System.out.println("Reject H0: mean significantly differs from 2.0");
} else {
    System.out.println("Fail to reject H0");
}

Two-Sample t-Test (Welch's)

Null hypothesis H₀: the two populations have equal means. Uses Welch's formula (unequal variances) by default.

java

double[] x = {2.5, 2.7, 2.3, 2.6, 2.4};
double[] y = {3.0, 3.2, 3.1, 2.9, 3.0};

TTest result = TTest.test(x, y, false); // false = unequal variances (Welch)
// or via facade:
TTest result = Hypothesis.t.test(x, y);

System.out.println(result);

Two-Sample t-Test (Equal Variance)

java

TTest result = TTest.test(x, y, true); // pooled variance
// or via facade:
TTest result = Hypothesis.t.test(x, y, "equal.var");

Use "equal.var" when you have reason to believe the two populations have the same variance (e.g. confirmed by an F-test).

Paired t-Test

Null hypothesis H₀: the mean difference between paired observations is zero. Suitable for before/after designs.

java

double[] before = {5.0, 5.5, 4.8, 5.2, 5.1};
double[] after  = {5.8, 6.0, 5.5, 5.7, 5.6};

TTest result = TTest.testPaired(before, after);
// or via facade:
TTest result = Hypothesis.t.test(before, after, "paired");

System.out.println(result);

Correlation t-Test

Tests whether the Pearson correlation coefficient differs significantly from 0.

java

double r = MathEx.cor(x, y);
int df = x.length - 2;

TTest result = TTest.test(r, df);
// or via facade:
TTest result = Hypothesis.t.test(r, df);

4. F-Tests and ANOVA

Two-Sample Variance Test

Null hypothesis H₀: the two populations have equal variances.

⚠️ The F-test is extremely sensitive to departures from normality. Use it only when the data is confirmed to be normally distributed.

java

double[] x = {1.0, 2.0, 3.0, 4.0, 5.0, 6.0};
double[] y = {1.1, 1.9, 2.8, 4.2, 5.1, 6.2};

FTest result = FTest.test(x, y);
// or via facade:
FTest result = Hypothesis.F.test(x, y);

System.out.println(result);
// → F-test(f = 1.1143, df1 = 5, df2 = 5, p-value = 0.9012)

System.out.printf("F=%.4f, df1=%d, df2=%d, p=%.4f%n",
    result.f(), result.df1(), result.df2(), result.pvalue());

One-Way ANOVA

Null hypothesis H₀: all group means are equal. Assumes equal variances across groups.

java

int[]    groups = {1,1,1,1,1,1, 2,2,2,2,2,2, 3,3,3,3,3,3};
double[] values = {6,8,4,5,3,4, 8,12,9,11,6,8, 13,9,11,8,7,12};

FTest result = FTest.test(groups, values);
// or via facade:
FTest result = Hypothesis.F.test(groups, values);

System.out.println(result);
// → F-test(f = 9.2647, df1 = 2, df2 = 15, p-value = 0.002399)

if (result.pvalue() < 0.05) {
    System.out.println("At least one group mean differs significantly.");
}

5. Kolmogorov-Smirnov Test

The KS test compares distributions via their empirical CDFs. It is non-parametric and sensitive to differences in both location and shape.

Note: Both test() overloads sort the input arrays in place. Pass copies if the original order must be preserved.

One-Sample KS Test

Null hypothesis H₀: the data was drawn from the reference distribution.

java

import smile.stat.distribution.GaussianDistribution;

double[] x = /* observed data */;

KSTest result = KSTest.test(x, GaussianDistribution.getInstance());
// or via facade:
KSTest result = Hypothesis.KS.test(x, GaussianDistribution.getInstance());

System.out.println(result);
// → One Sample Kolmogorov-Smirnov Test(d = 0.0930, p-value = 0.7598)

System.out.printf("D=%.4f, p=%.4f%n", result.d(), result.pvalue());
// Large p-value → fail to reject H0 (data is consistent with N(0,1))

Two-Sample KS Test

Null hypothesis H₀: both samples are drawn from the same distribution.

java

double[] x = /* sample 1 */;
double[] y = /* sample 2 */;

KSTest result = KSTest.test(x, y);
// or via facade:
KSTest result = Hypothesis.KS.test(x, y);

System.out.println(result);
// Small p-value → distributions differ significantly

6. Correlation Tests

CorTest provides three correlation measures, each with an associated significance test.

Pearson Correlation

Measures linear association. Assumes bivariate normality.

java

double[] x = {44.4, 45.9, 41.9, 53.3, 44.7, 44.1, 50.7, 45.2, 60.1};
double[] y = { 2.6,  3.1,  2.5,  5.0,  3.6,  4.0,  5.2,  2.8,  3.8};

CorTest result = CorTest.pearson(x, y);
// or via facade:
CorTest result = Hypothesis.cor.test(x, y);           // default = Pearson
CorTest result = Hypothesis.cor.test(x, y, "pearson");

System.out.println(result);
// → Pearson Correlation Test(cor = 0.57, t = 1.8411, df = 7, p-value = 0.1082)

System.out.printf("r=%.4f, t=%.4f, df=%.0f, p=%.4f%n",
    result.cor(), result.t(), result.df(), result.pvalue());

Spearman Rank Correlation

Measures monotone association. Distribution-free (non-parametric).

java

CorTest result = CorTest.spearman(x, y);
// or:
CorTest result = Hypothesis.cor.test(x, y, "spearman");

System.out.printf("ρ=%.4f, p=%.4f%n", result.cor(), result.pvalue());

Kendall Rank Correlation

Measures concordance between rankings. More robust than Spearman for small samples with many ties.

java

CorTest result = CorTest.kendall(x, y);
// or:
CorTest result = Hypothesis.cor.test(x, y, "kendall");

System.out.printf("τ=%.4f, p=%.4f%n", result.cor(), result.pvalue());

Note: The Kendall test is non-parametric. Its df() field is set to 0. The p-value uses a normal approximation valid for n ≥ 10.

Choosing a Correlation Method

Scenario	Recommended method
Continuous, normally distributed data	Pearson
Ordinal data, or non-normal continuous	Spearman
Small samples with ties	Kendall

7. Chi-Square Tests

Pearson's chi-square tests compare observed counts to expected counts.

⚠️ The chi-square approximation is only valid when expected counts are reasonably large (typically ≥ 5 per cell). With small counts, consider Fisher's exact test.

One-Sample Goodness-of-Fit

Null hypothesis H₀: the observed frequencies follow the expected distribution.

java

// Observed die rolls
int[]    bins = {20, 22, 13, 22, 10, 13};
double[] prob = {1.0/6, 1.0/6, 1.0/6, 1.0/6, 1.0/6, 1.0/6}; // fair die

ChiSqTest result = ChiSqTest.test(bins, prob);
// or via facade:
ChiSqTest result = Hypothesis.chisq.test(bins, prob);

System.out.println(result);
// → One Sample Chi-squared Test(t = 8.36, df = 5, p-value = 0.1370)

System.out.printf("χ²=%.4f, df=%.0f, p=%.4f%n",
    result.chisq(), result.df(), result.pvalue());

Specify a custom number of constraints (default = 1) when parameters were estimated from the data:

java

// Constraints = 1 (default): one constraint from the total count
ChiSqTest result = ChiSqTest.test(bins, prob, 1);
// or:
ChiSqTest result = Hypothesis.chisq.test(bins, prob, 1);

Two-Sample Chi-Square Test

Null hypothesis H₀: both samples have the same underlying distribution.

java

int[] bins1 = {8, 13, 16, 10, 3};
int[] bins2 = {4,  9, 14, 16, 7};

ChiSqTest result = ChiSqTest.test(bins1, bins2);
// or via facade:
ChiSqTest result = Hypothesis.chisq.test(bins1, bins2);

System.out.printf("χ²=%.4f, df=%.0f, p=%.4f%n",
    result.chisq(), result.df(), result.pvalue());

Chi-Square Test of Independence (Contingency Table)

Null hypothesis H₀: the two categorical variables are independent.

Also computes Cramér's V — a measure of association strength between 0 (no association) and 1 (perfect association). For 2×2 tables, equals the Phi coefficient; continuity correction is applied automatically.

java

// 2×2 contingency table
int[][] table = {
    {12,  7},
    { 5,  7}
};

ChiSqTest result = ChiSqTest.test(table);
// or via facade:
ChiSqTest result = Hypothesis.chisq.test(table);

System.out.println(result);
// → Contingency Table Chi-squared Test(t = 0.9..., df = 1, p-value = 0.4233, Cramer's V = 0.22)

System.out.printf("χ²=%.4f, df=%.0f, p=%.4f, V=%.4f%n",
    result.chisq(), result.df(), result.pvalue(), result.CramerV());

Larger tables (r × c) work the same way:

java

int[][] table3x3 = {
    {50, 30, 20},
    {15, 45, 40},
    {35, 25, 40}
};
ChiSqTest result = ChiSqTest.test(table3x3);

8. Interpreting Results

The p-value

The p-value is the probability of observing data at least as extreme as the sample, assuming the null hypothesis is true. A small p-value provides evidence against H₀.

p-value	Interpretation
< 0.001	Very strong evidence against H₀
0.001–0.01	Strong evidence against H₀
0.01–0.05	Moderate evidence against H₀
0.05–0.1	Weak / suggestive evidence
≥ 0.1	Insufficient evidence to reject H₀

Common misconception: a large p-value does not "prove" H₀. It merely means the data are consistent with H₀.

Two-Sided vs. One-Sided Tests

All tests in this package report two-sided p-values. To obtain a one-sided p-value for a t-test, divide by 2 (for symmetric distributions):

java

TTest t = TTest.test(x, 0.0);
double oneSidedP = t.pvalue() / 2.0; // if you predicted the direction

Multiple Comparisons

If you run many tests simultaneously, consider correcting for multiple comparisons (Bonferroni, Benjamini-Hochberg, etc.) to control the family-wise error rate or false discovery rate.

Effect Size

The p-value tells you whether an effect exists, not how large it is. Always report an effect size alongside the p-value:

Test	Suggested effect size
t-test	Cohen's d
F-test / ANOVA	η² (eta-squared)
Correlation	r (Pearson), ρ (Spearman), τ (Kendall)
Chi-square	Cramér's V (provided in `CramerV()`)

9. Quick Reference

java

import smile.stat.Hypothesis;
import smile.stat.hypothesis.*;

// ── Significance code ─────────────────────────────────────────────
Hypothesis.significance(0.003); // → "**"

// ── t-Tests ───────────────────────────────────────────────────────
TTest r = Hypothesis.t.test(x, mu0);          // one-sample
TTest r = Hypothesis.t.test(x, y);            // Welch two-sample
TTest r = Hypothesis.t.test(x, y, "equal.var"); // pooled two-sample
TTest r = Hypothesis.t.test(x, y, "paired");  // paired
TTest r = Hypothesis.t.test(corr, df);        // correlation significance

r.t();       // t-statistic
r.df();      // degrees of freedom
r.pvalue();  // two-sided p-value

// ── F-Test / ANOVA ────────────────────────────────────────────────
FTest r = Hypothesis.F.test(x, y);            // variance equality
FTest r = Hypothesis.F.test(labels, values);  // one-way ANOVA

r.f();       // F-statistic
r.df1();     // numerator df
r.df2();     // denominator df
r.pvalue();

// ── KS Test ───────────────────────────────────────────────────────
KSTest r = Hypothesis.KS.test(x, distribution); // one-sample
KSTest r = Hypothesis.KS.test(x, y);            // two-sample

r.d();       // KS statistic (max |F_n - F|)
r.pvalue();

// ── Correlation Tests ─────────────────────────────────────────────
CorTest r = Hypothesis.cor.test(x, y);              // Pearson
CorTest r = Hypothesis.cor.test(x, y, "spearman");  // Spearman
CorTest r = Hypothesis.cor.test(x, y, "kendall");   // Kendall

r.cor();     // correlation coefficient
r.t();       // test statistic
r.df();      // degrees of freedom (0 for Kendall)
r.pvalue();

// ── Chi-Square Tests ──────────────────────────────────────────────
ChiSqTest r = Hypothesis.chisq.test(bins, prob);         // goodness-of-fit
ChiSqTest r = Hypothesis.chisq.test(bins, prob, k);      // with k constraints
ChiSqTest r = Hypothesis.chisq.test(bins1, bins2);       // two-sample
ChiSqTest r = Hypothesis.chisq.test(bins1, bins2, k);    // with k constraints
ChiSqTest r = Hypothesis.chisq.test(contingencyTable);   // independence

r.chisq();   // chi-square statistic
r.df();      // degrees of freedom
r.pvalue();
r.CramerV(); // effect size (contingency table only)

Direct API (without facade)

java

TTest.test(x, mean);
TTest.test(x, y, equalVariance);
TTest.testPaired(x, y);
TTest.test(r, df);

FTest.test(x, y);
FTest.test(labels, values);          // ANOVA

KSTest.test(x, distribution);
KSTest.test(x, y);

CorTest.pearson(x, y);
CorTest.spearman(x, y);
CorTest.kendall(x, y);

ChiSqTest.test(bins, prob);
ChiSqTest.test(bins, prob, constraints);
ChiSqTest.test(bins1, bins2);
ChiSqTest.test(bins1, bins2, constraints);
ChiSqTest.test(table);

SMILE — Hypothesis Testing

SMILE — Hypothesis Testing

Table of Contents

1. Overview

Significance Codes

2. The Hypothesis Facade

3. t-Tests

One-Sample t-Test

Two-Sample t-Test (Welch's)

Two-Sample t-Test (Equal Variance)

Paired t-Test

Correlation t-Test

4. F-Tests and ANOVA

Two-Sample Variance Test

One-Way ANOVA

5. Kolmogorov-Smirnov Test

One-Sample KS Test

Two-Sample KS Test

6. Correlation Tests

Pearson Correlation

Spearman Rank Correlation

Kendall Rank Correlation

Choosing a Correlation Method

7. Chi-Square Tests

One-Sample Goodness-of-Fit

Two-Sample Chi-Square Test

Chi-Square Test of Independence (Contingency Table)

8. Interpreting Results

The p-value

Two-Sided vs. One-Sided Tests

Multiple Comparisons

Effect Size

9. Quick Reference

Direct API (without facade)

2. The `Hypothesis` Facade