SPSS command: SPSS HILOGLINEAR

Hierarchical loglinear analysis of multi-way tables

Q: What is a generating class?

The highest-order terms of a hierarchical model. A generating class of age*device, device*bought implies the main effects of all three variables and those two associations, but no age*bought term.

Q: Why are expected counts fitted to full convergence?

Iterative proportional fitting here runs until the expected counts stop changing, rather than stopping after SPSS's default 20 cycles, so they are exact maximum-likelihood values. Results can differ from SPSS in the last decimals when SPSS stopped early.

Q: How do I analyse a table I only have as counts?

Enter one row per cell with a count column, set it as the workspace weight and declare it a frequency weight. Each row then counts as that many identical cases.

Q: Why are partial associations missing?

They and the parameter estimates are computed only when Model is Saturated. Start from the saturated model and let backward elimination simplify it.

Hierarchical loglinear analysis looks for the simplest set of associations that reproduces a cross-tabulation of two to six categorical variables.

Try with example data Run on your own file

Free · No account · Files stay on your device · Updated 18 October 2026

Worked example result

Partial Associations · age

Partial Chi-Square(1) = 0.250

p = .617

Effect	df	Partial Chi-Square	Sig.
age	1	0.250	.617
device	1	1.962	.161
bought	1	41.689	<.001
age*device	1	53.699	<.001
age*bought	1	0.018	.892

Partial Associations

See the full worked example

What it tells you

Crosstabs runs the SPSS HILOGLINEAR procedure on your device: it tests whole orders of interaction (K-way effects), measures each effect's partial association, and removes unnecessary terms by backward elimination until the remaining generating class still fits the table.

Whether all effects of a given order — all two-way associations, the three-way interaction — can be dropped together (K-Way and Higher-Order Effects).
How much each individual association contributes after adjusting for the others (Partial Associations).
The smallest hierarchical model that still fits, step by step (Step Summary), and its likelihood-ratio and Pearson fit.
Which cells the final model misses, from observed and expected counts with residuals.

When to use it

Use it when

Three or more categorical variables and you want to know which pairs are associated after accounting for the rest.
Checking for conditional independence, such as whether age and purchase are unrelated once device is known.
Survey or audit tables where no variable is singled out as the outcome.

Use something else when

Only two variables: a chi-square test in Crosstabs answers the same question directly.
One variable is clearly the response: a logit model in general loglinear analysis or logistic regression is easier to interpret.
Many sparse cells, such as six factors with few cases: expected counts become tiny and chi-square tests unreliable.

How to run it in Crosstabs

01
Choose Hierarchical loglinear analysis
Open case-level data in Statistics mode and, in the General linear model group, pick Hierarchical loglinear analysis (HILOGLINEAR). For an aggregated table, declare the count column as a frequency weight.
02
Tick the factors
Under Factors (2 to 6) tick the categorical variables that form the table.
03
Pick the starting model
Set Model to Saturated (needed for partial associations and parameter estimates), All two-way interactions or Main effects only.
04
Backward elimination and run
Keep Use backward elimination ticked to drop effects whose removal p exceeds .05, then click Run.

Run Hierarchical Loglinear Analysis in your browser

Worked example

Computed by the Crosstabs engine from the synthetic data below when this page was built — not typed in.

Try with example data

Data

400 synthetic shoppers cross-classified by age (Under 35, 35+), device (Mobile, Desktop) and bought (Yes, No). Younger shoppers use mobile more, and mobile shoppers buy less often; age has no direct link to buying (fixed seed). Model = Saturated with Use backward elimination ticked.

K-Way and Higher-Order Effects

Item	K	df	Likelihood Ratio Chi-Square	Likelihood Ratio Sig.	Pearson Chi-Square	Pearson Sig.	Number of Iterations
K-way and Higher Order Effects	1	7	104.405	<.001	110.400	<.001	0
	2	4	60.504	<.001	59.901	<.001	2
	3	1	0.002	.967	0.002	.967	12
K-way Effects	1	3	43.901	<.001	50.499	<.001	0
	2	3	60.502	<.001	59.899	<.001	0
	3	1	0.002	.967	0.002	.967	0

Partial Associations

Effect	df	Partial Chi-Square	Sig.	Number of Iterations
age	1	0.250	.617	2
device	1	1.962	.161	2
bought	1	41.689	<.001	2
age*device	1	53.699	<.001	2
age*bought	1	0.018	.892	2
device*bought	1	5.600	.018	2

Goodness-of-Fit Tests

Item	Chi-Square	df	Sig.
Likelihood Ratio	0.020	2	.990
Pearson	0.020	2	.990

Model (generating class): age*device, device*bought.

How to read it

The three-way interaction is negligible (K = 3: likelihood-ratio χ²(1) = 0.002, p = .967). Partial associations show significant age×device (χ² = 53.70, p < .001) and device×bought (χ² = 5.60, p = .018) links but no age×bought link (p = .892). Backward elimination keeps exactly those two generating classes: the final model fits with χ²(2) = 0.020, p = .990 — age and buying are independent once device is known.

Output tables explained

Data Information: Valid, missing and weighted cases, and the number of categories of each factor.
K-Way and Higher-Order Effects: Likelihood-ratio and Pearson tests that all effects of order K and higher are zero, and that the K-way effects alone are zero.
Partial Associations: For each effect, the partial chi-square from removing it from the model of its order; shown for a saturated starting model.
Parameter Estimates: Saturated-model effect-coded (sum-to-zero) estimates with 0.5 added to every cell, with Z tests and confidence intervals.
Step Summary: Each elimination step: the generating class, the effect deleted and its chi-square change.
Goodness-of-Fit Tests: Likelihood-ratio and Pearson chi-square of the final model; a large p means the model reproduces the table. Cell Counts and Residuals follow.

Assumptions and notes

Cell counts are independent Poisson or multinomial counts — each case appears in one cell only.
Hierarchy: whenever an interaction is in the model, all of its lower-order terms are too.
Expected counts are not too small; most should exceed 5 for the chi-square approximations.

How the results are validated

HILOGLINEAR is checked against Poisson GLM fits in statsmodels and a NumPy iterative-proportional-fitting implementation on a four-factor table: K-way likelihood-ratio and Pearson statistics and partial associations agree within 1e-7, saturated effect-coded estimates and standard errors (delta 0.5) within 1e-8, and backward elimination must reach the same final model, with its fit statistics within 1e-7 and identical df. Validation evidence

Questions

What is a generating class?

The highest-order terms of a hierarchical model. A generating class of age*device, device*bought implies the main effects of all three variables and those two associations, but no age*bought term.

Why are expected counts fitted to full convergence?

Iterative proportional fitting here runs until the expected counts stop changing, rather than stopping after SPSS's default 20 cycles, so they are exact maximum-likelihood values. Results can differ from SPSS in the last decimals when SPSS stopped early.

How do I analyse a table I only have as counts?

Enter one row per cell with a count column, set it as the workspace weight and declare it a frequency weight. Each row then counts as that many identical cases.

Why are partial associations missing?

They and the parameter estimates are computed only when Model is Saturated. Start from the saturated model and let backward elimination simplify it.

Run Hierarchical Loglinear Analysis on your own data

Open a .sav, CSV or Excel file. Everything runs in your browser; the file never leaves your device.

Open the workspace

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