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Test Results

All hypothesis test functions return a TestResult object: 
type TestResult = {
  statistic: number;  // Test statistic value
  pvalue: number;     // Probability of observing this result under null hypothesis
};
Interpreting p-values:
  • p < 0.05: Statistically significant at 5% level (reject null hypothesis)
  • p < 0.01: Statistically significant at 1% level (strong evidence)
  • p ≥ 0.05: Not statistically significant (fail to reject null hypothesis)

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t-Tests

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ttest_1samp

One-sample t-test. 
function ttest_1samp(
  a: Tensor,
  popmean: number
): TestResult
​
a
Tensor
required
Sample data
​
popmean
number
required
Expected population mean under null hypothesis
​
return
TestResult
Test result with statistic and p-value
Tests whether the mean of a sample differs from a known population mean. Null Hypothesis (H₀): The sample mean equals the population mean (μ = μ₀) Statistical Context: The test statistic is t = (x̄ - μ₀) / (s / √n) where x̄ is sample mean, s is sample standard deviation, and n is sample size. Under H₀, this follows a t-distribution with n-1 degrees of freedom. Assumptions:
  • Sample is randomly drawn from population
  • Data is approximately normally distributed (or n > 30 by CLT)
  • Observations are independent
Requires at least 2 samples. 
const sample = tensor([5.2, 4.8, 5.1, 5.3, 4.9]);
const result = ttest_1samp(sample, 5.0);
// Test if sample mean differs from 5.0

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ttest_ind

Independent two-sample t-test. 
function ttest_ind(
  a: Tensor,
  b: Tensor,
  equalVar?: boolean
): TestResult
​
a
Tensor
required
First sample
​
b
Tensor
required
Second sample
​
equalVar
boolean
default:"true"
If true, uses pooled variance (Student’s t-test). If false, uses Welch’s t-test (unequal variances).
​
return
TestResult
Test result with statistic and p-value
Tests whether means of two independent samples are equal. Null Hypothesis (H₀): The two population means are equal (μ₁ = μ₂) Statistical Context:
  • Student’s t-test (equalVar=true): Assumes equal population variances, uses pooled variance estimate
  • Welch’s t-test (equalVar=false): Does not assume equal variances, uses Welch-Satterthwaite approximation for degrees of freedom
Assumptions:
  • Samples are independent
  • Data in each group is approximately normally distributed
  • If equalVar=true, population variances are equal (test with Levene’s test)
Requires at least 2 samples in each group. 
const control = tensor([4.2, 4.5, 4.3, 4.6, 4.4]);
const treatment = tensor([5.1, 5.3, 5.2, 5.4, 5.0]);
const result = ttest_ind(control, treatment);
// Test if treatment group has different mean than control

​
ttest_rel

Paired-sample t-test. 
function ttest_rel(
  a: Tensor,
  b: Tensor
): TestResult
​
a
Tensor
required
First set of measurements (before)
​
b
Tensor
required
Second set of measurements (after) - must have same length as a
​
return
TestResult
Test result with statistic and p-value
Tests whether means of two related samples are equal. Null Hypothesis (H₀): The mean difference is zero (μ_diff = 0) Statistical Context: The paired t-test is equivalent to a one-sample t-test on the differences. It’s more powerful than the independent t-test when observations are paired because it controls for individual variation. Use Cases:
  • Before/after measurements on same subjects
  • Matched pairs (twins, siblings)
  • Repeated measurements on same experimental units
  • Left vs right measurements on same subjects
Assumptions:
  • Pairs are randomly selected
  • Differences are approximately normally distributed
  • Pairs are independent of each other
Requires at least 2 paired samples. 
const before = tensor([120, 135, 140, 130, 125]);
const after = tensor([115, 128, 135, 125, 120]);
const result = ttest_rel(before, after);
// Test if blood pressure changed after treatment

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ANOVA

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f_oneway

One-way ANOVA. 
function f_oneway(
  ...samples: Tensor[]
): TestResult
​
samples
Tensor[]
required
Two or more sample groups to compare
​
return
TestResult
Test result with F-statistic and p-value
Tests whether means of two or more groups are all equal. Null Hypothesis (H₀): All group means are equal (μ₁ = μ₂ = … = μₖ) Statistical Context: ANOVA (Analysis of Variance) decomposes total variance into between-group and within-group components. The F-statistic is the ratio of between-group variance to within-group variance. Large F values indicate greater differences between groups relative to within-group variation. Assumptions:
  • Samples are independent
  • Data in each group is approximately normally distributed
  • Population variances are equal (homoscedasticity - test with Levene’s test)
Post-hoc Tests: If ANOVA is significant, use post-hoc tests (like Tukey’s HSD) to determine which specific groups differ. 
const group1 = tensor([23, 25, 24, 26, 25]);
const group2 = tensor([28, 30, 29, 31, 30]);
const group3 = tensor([20, 22, 21, 23, 22]);
const result = f_oneway(group1, group2, group3);
// Test if any group means differ

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Variance Tests

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levene

Levene’s test for equality of variances. 
function levene(
  center: 'mean' | 'median' | 'trimmed',
  ...samples: Tensor[]
): TestResult
​
center
'mean' | 'median' | 'trimmed'
required
Method for centering:
  • 'median': Most robust to non-normality (recommended)
  • 'mean': Traditional Levene’s test
  • 'trimmed': 10% trimmed mean (compromise)
​
samples
Tensor[]
required
Two or more sample groups to compare (each group must have at least 2 samples)
​
return
TestResult
Test result with W-statistic and p-value
Tests whether two or more groups have equal variances. More robust than Bartlett’s test for non-normal data. Null Hypothesis (H₀): All group variances are equal (σ₁² = σ₂² = … = σₖ²) Statistical Context: Levene’s test performs an ANOVA on absolute deviations from group centers. The ‘median’ option (Brown-Forsythe test) is most robust to departures from normality. Use Cases:
  • Testing ANOVA assumption of homogeneity of variance
  • Determining whether to use pooled or unpooled variance in t-tests
  • Quality control (comparing process variability)
const group1 = tensor([1, 2, 3, 4, 5]);
const group2 = tensor([2, 4, 6, 8, 10]);
const result = levene('median', group1, group2);

​
bartlett

Bartlett’s test for equality of variances. 
function bartlett(
  ...samples: Tensor[]
): TestResult
​
samples
Tensor[]
required
Two or more sample groups to compare (each group must have at least 2 samples)
​
return
TestResult
Test result with chi-square statistic and p-value
Tests whether two or more groups have equal variances. Assumes data is normally distributed; use Levene’s test for non-normal data. Null Hypothesis (H₀): All group variances are equal (σ₁² = σ₂² = … = σₖ²) Statistical Context: Bartlett’s test is more powerful than Levene’s test when data is normally distributed, but very sensitive to departures from normality. The test statistic approximately follows a chi-square distribution with k-1 degrees of freedom. When to Use:
  • Data is known to be normally distributed
  • Need maximum power for detecting variance differences
  • For non-normal data, use Levene’s test instead
const group1 = tensor([1, 2, 3, 4, 5]);
const group2 = tensor([2, 4, 6, 8, 10]);
const result = bartlett(group1, group2);

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Chi-Square Tests

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chisquare

Chi-square goodness of fit test. 
function chisquare(
  f_obs: Tensor,
  f_exp?: Tensor
): TestResult
​
f_obs
Tensor
required
Observed frequencies (must be non-negative)
​
f_exp
Tensor
Expected frequencies (must be positive and sum to same total as f_obs). If not provided, assumes uniform distribution.
​
return
TestResult
Test result with chi-square statistic and p-value
Tests whether observed frequencies differ from expected frequencies. Null Hypothesis (H₀): Observed frequencies follow the expected distribution Statistical Context: The test statistic is χ² = Σ((O - E)² / E) where O is observed and E is expected frequency. Under H₀, this follows a chi-square distribution with k-1 degrees of freedom. Assumptions:
  • Categories are mutually exclusive
  • Observations are independent
  • Expected frequency in each category should be at least 5 (rule of thumb)
Use Cases:
  • Testing goodness of fit to theoretical distribution
  • Testing uniformity of categorical data
  • Comparing observed vs expected counts
// Test if die is fair (expected uniform distribution)
const rolls = tensor([18, 22, 19, 21, 20, 20]); // 120 rolls
const result = chisquare(rolls);

// Test specific distribution
const observed = tensor([30, 45, 25]);
const expected = tensor([25, 50, 25]);
const result2 = chisquare(observed, expected);

​
Normality Tests

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shapiro

Shapiro-Wilk test for normality. 
function shapiro(
  x: Tensor
): TestResult
​
x
Tensor
required
Sample data (size must be between 3 and 5000)
​
return
TestResult
Test result with W-statistic and p-value
Tests whether data comes from a normal distribution. Null Hypothesis (H₀): The data is normally distributed Statistical Context: The Shapiro-Wilk test is based on the correlation between the data and the corresponding normal scores. W-statistic ranges from 0 to 1, with values closer to 1 indicating normality. This implementation uses Algorithm AS R94 (Royston, 1995). Power: Shapiro-Wilk is one of the most powerful normality tests, especially for small to moderate sample sizes. It’s generally preferred over Kolmogorov-Smirnov. Sample Size:
  • Minimum: 3 observations
  • Maximum: 5000 observations
  • Most powerful for n < 50
const data = tensor([2.3, 2.5, 2.4, 2.6, 2.5, 2.7]);
const result = shapiro(data);
// Test if data is normally distributed

​
normaltest

D’Agostino-Pearson omnibus test for normality. 
function normaltest(
  a: Tensor
): TestResult
​
a
Tensor
required
Sample data (requires at least 8 samples)
​
return
TestResult
Test result with K² statistic and p-value
Tests for normality using D’Agostino-Pearson’s omnibus test combining skewness and kurtosis. Null Hypothesis (H₀): The data is normally distributed Statistical Context: Combines tests for skewness and kurtosis into a single omnibus test. The test statistic K² = Z₁² + Z₂² approximately follows a chi-square distribution with 2 degrees of freedom, where Z₁ is the standardized skewness and Z₂ is the standardized kurtosis. Advantages:
  • Works well for larger sample sizes
  • Explicitly tests both skewness and kurtosis
  • Less affected by ties than Shapiro-Wilk
Requires at least 8 samples. 
const data = tensor([1.2, 2.3, 1.9, 2.5, 2.1, 2.8, 2.0, 2.4]);
const result = normaltest(data);

​
anderson

Anderson-Darling test for normality. 
function anderson(
  x: Tensor
): {
  statistic: number;
  critical_values: number[];
  significance_level: number[];
}
​
x
Tensor
required
Sample data
​
return
object
Object with:
  • statistic: Anderson-Darling A² statistic
  • critical_values: Critical values at [15%, 10%, 5%, 2.5%, 1%] significance levels
  • significance_level: [0.15, 0.1, 0.05, 0.025, 0.01]
Tests for normality using the Anderson-Darling test. Null Hypothesis (H₀): The data is normally distributed Statistical Context: The Anderson-Darling test is a modification of the Kolmogorov-Smirnov test that gives more weight to the tails. It’s particularly sensitive to departures from normality in the tails of the distribution. Interpretation: If the statistic is larger than the critical value at a given significance level, reject the null hypothesis at that level. Use Cases:
  • Testing normality with emphasis on tail behavior
  • When outliers or tail behavior is important
  • Quality control applications
const data = tensor([1.2, 2.3, 1.9, 2.5, 2.1]);
const result = anderson(data);
// Compare statistic to critical_values
if (result.statistic > result.critical_values[2]) {
  console.log('Reject normality at 5% level');
}

​
kstest

Kolmogorov-Smirnov test for goodness of fit. 
function kstest(
  data: Tensor,
  cdf: 'norm' | ((x: number) => number)
): TestResult
​
data
Tensor
required
Sample data
​
cdf
'norm' | ((x: number) => number)
required
Cumulative distribution function to test against. Use 'norm' for standard normal, or provide custom CDF function.
​
return
TestResult
Test result with D-statistic and p-value
Tests whether data comes from a specified distribution. Null Hypothesis (H₀): The data follows the specified distribution Statistical Context: The KS test computes the maximum vertical distance (D-statistic) between the empirical CDF and the theoretical CDF. The p-value uses the Kolmogorov distribution (asymptotic approximation). Characteristics:
  • Distribution-free (non-parametric)
  • Works with continuous distributions
  • Less powerful than Shapiro-Wilk for testing normality
  • More general (can test any distribution)
// Test against standard normal
const data = tensor([0.1, -0.5, 0.8, -0.2, 0.3]);
const result = kstest(data, 'norm');

// Test against custom distribution
const customCdf = (x: number) => 1 / (1 + Math.exp(-x)); // Logistic CDF
const result2 = kstest(data, customCdf);

​
Non-Parametric Tests

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mannwhitneyu

Mann-Whitney U test (non-parametric). 
function mannwhitneyu(
  x: Tensor,
  y: Tensor
): TestResult
​
x
Tensor
required
First sample
​
y
Tensor
required
Second sample
​
return
TestResult
Test result with U-statistic and p-value (using normal approximation with tie and continuity correction)
Tests whether two independent samples come from the same distribution. Null Hypothesis (H₀): The two populations have the same distribution Statistical Context: The Mann-Whitney U test (also called Wilcoxon rank-sum test) is the non-parametric alternative to the independent t-test. It ranks all observations from both groups together and tests whether the rank sums differ significantly. Advantages over t-test:
  • No assumption of normality
  • Robust to outliers
  • Works with ordinal data
  • More powerful than t-test when assumptions are violated
When to Use:
  • Non-normal distributions
  • Ordinal data
  • Presence of outliers
  • Small sample sizes
Uses normal approximation with tie and continuity correction. 
const control = tensor([12, 15, 14, 13, 16]);
const treatment = tensor([18, 20, 19, 21, 17]);
const result = mannwhitneyu(control, treatment);

​
wilcoxon

Wilcoxon signed-rank test (non-parametric paired test). 
function wilcoxon(
  x: Tensor,
  y?: Tensor
): TestResult
​
x
Tensor
required
First set of measurements (or differences if y not provided)
​
y
Tensor
Second set of measurements. If provided, tests paired differences.
​
return
TestResult
Test result with W+ statistic and p-value (using normal approximation with tie and continuity correction)
Non-parametric test for paired samples or single sample median. Null Hypothesis (H₀): The median of differences is zero Statistical Context: The Wilcoxon signed-rank test is the non-parametric alternative to the paired t-test. It ranks the absolute values of differences and tests whether positive ranks dominate negative ranks (or vice versa). Advantages over paired t-test:
  • No assumption of normality for differences
  • Robust to outliers
  • More powerful when assumptions are violated
When to Use:
  • Non-normal differences
  • Ordinal data
  • Presence of outliers in differences
  • Small sample sizes
Uses normal approximation with tie and continuity correction. Zero differences are excluded. 
// Paired test
const before = tensor([85, 90, 88, 92, 87]);
const after = tensor([80, 85, 83, 88, 84]);
const result = wilcoxon(before, after);

// One-sample test (test if median differs from 0)
const diffs = tensor([2.5, -1.0, 3.2, 1.5, -0.5]);
const result2 = wilcoxon(diffs);

​
kruskal

Kruskal-Wallis H-test (non-parametric version of ANOVA). 
function kruskal(
  ...samples: Tensor[]
): TestResult
​
samples
Tensor[]
required
Two or more sample groups to compare
​
return
TestResult
Test result with H-statistic and p-value (using chi-square approximation with tie correction)
Tests whether samples come from the same distribution (non-parametric alternative to one-way ANOVA). Null Hypothesis (H₀): All groups have the same distribution Statistical Context: The Kruskal-Wallis test ranks all observations across all groups and tests whether the mean ranks differ between groups. The H-statistic approximately follows a chi-square distribution with k-1 degrees of freedom. Advantages over ANOVA:
  • No assumption of normality
  • No assumption of equal variances
  • Robust to outliers
  • Works with ordinal data
Post-hoc Tests: If significant, use Dunn’s test or pairwise Mann-Whitney tests with multiple comparison correction. Uses chi-square approximation with tie correction. 
const group1 = tensor([23, 25, 24, 26, 25]);
const group2 = tensor([28, 30, 29, 31, 30]);
const group3 = tensor([20, 22, 21, 23, 22]);
const result = kruskal(group1, group2, group3);

​
friedmanchisquare

Friedman test (non-parametric repeated measures ANOVA). 
function friedmanchisquare(
  ...samples: Tensor[]
): TestResult
​
samples
Tensor[]
required
Three or more related samples (all must have the same length)
​
return
TestResult
Test result with chi-square statistic and p-value (using chi-square approximation with tie correction)
Tests whether related samples have different distributions (non-parametric alternative to repeated measures ANOVA). Null Hypothesis (H₀): The distributions of all related samples are the same Statistical Context: The Friedman test ranks observations within each block (subject) and tests whether the mean ranks differ across treatments. It’s used when the same subjects are measured under different conditions. Use Cases:
  • Repeated measurements on same subjects
  • Matched groups (e.g., siblings, matched controls)
  • Block designs in experiments
  • When data is ordinal or non-normal
Assumptions:
  • Observations are ranked within blocks
  • Blocks are independent
  • At least 3 related samples required
Uses chi-square approximation with tie correction. 
// 5 subjects tested under 3 conditions
const condition1 = tensor([8, 9, 7, 8, 9]);
const condition2 = tensor([9, 10, 8, 9, 10]);
const condition3 = tensor([7, 8, 6, 7, 8]);
const result = friedmanchisquare(condition1, condition2, condition3);

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