Quick Answer: Fixed Effects Regression Result

The verified workbook contains 649 complete observations. The outcome is G3. Numeric predictors are G1, G2, studytime, failures, absences, age, Medu and Fedu. The multi-way fixed-effects specification adds indicator variables for school, sex, address and higher, with GP, F, R and no used as the reference levels implied by the coefficient labels.

The multi-way model has R² = 0.852652, adjusted R² = 0.849872, RMSE = 1.239162 and MAE = 0.777422. It has the highest adjusted R² among the three displayed models. However, its nested improvement over pooled OLS is not statistically significant at alpha .05, F = 2.022938, df difference = 4, p = 0.089644.

What Is Fixed Effects Regression?

Fixed Effects Regression controls for group-specific intercept differences that might otherwise be absorbed into the error term and confound the coefficients of interest. In a least-squares dummy-variable model, each nonreference group receives an indicator coefficient. In a within-transformed panel model, group means are subtracted from the outcome and time-varying predictors so that estimation uses variation within the same entity.

The term “fixed effect” can be used in several ways. In experimental ANOVA, a fixed factor has levels chosen because those exact levels are substantively important. In regression, fixed effects often mean a complete set of group indicators. In panel econometrics, entity fixed effects control all time-invariant characteristics of each entity, whether those characteristics were measured or not.

What Fixed Effects Control

A group dummy controls for average level differences between that group and the reference group. In repeated panel data, an entity fixed effect controls all characteristics that remain constant within the entity across time. This can reduce omitted-variable bias from stable unmeasured differences.

What Fixed Effects Do Not Automatically Control

Fixed effects do not remove bias from time-varying omitted variables, measurement error, reverse causation, incorrect functional form or selection mechanisms. Adding many group dummies does not transform an observational association into a causal effect without a defensible research design.

One-Way, Two-Way and Multi-Way Fixed Effects

The workbook compares pooled OLS with a one-way school fixed-effects model and a multi-way model containing four grouping dimensions. The one-way model has nine slope degrees of freedom: eight numeric predictors plus one school dummy. The multi-way model has twelve slope degrees of freedom: eight numeric predictors plus four group dummies.

Fixed Effects Regression Equations and Within Transformation

Least-Squares Dummy-Variable Equation

The intercept represents the expected G3 for the reference categories when numeric predictors equal zero. Each numeric coefficient is interpreted while holding the other numeric predictors and fixed-effect dummies constant. Each dummy coefficient represents an adjusted intercept difference from its reference group.

Panel Fixed-Effects Equation

Here, αᵢ is an entity fixed effect and λₜ may be a time fixed effect. Repeated observations are required to estimate within-entity change. Time-invariant predictors cannot be separately identified from entity fixed effects because they disappear under demeaning.

Within Transformation

Subtracting each entity’s mean removes the entity-specific intercept. In the uploaded workbook’s chart demonstration, G2 and G3 are demeaned within the combined school × sex × address × higher cells. The within-transformed G2–G3 correlation is 0.902094, and the simple within slope is approximately 0.987075.

When to Use Fixed Effects Regression

Fixed effects are particularly useful when omitted stable group characteristics are likely to correlate with the predictors. They are less useful when the main predictors have almost no within-group variation or when the group indicators consume too much information relative to sample size.

Dataset and Variables Used

The worked analysis uses 649 complete student records. The outcome is G3. Eight numeric predictors enter all three models. The one-way model adds school, while the multi-way model adds school, sex, address and higher-education intention.

Outcome and Numeric Predictor Dictionary

Fixed-Effect Group Dictionary and Reference Categories

Complete Model Specifications

Pooled OLS vs One-Way and Multi-Way Fixed Effects

The multi-way model increases R² by 0.001875 over pooled OLS and increases adjusted R² by 0.000960. RMSE decreases by approximately 0.007858, while MAE decreases by about 0.002596. These are real improvements but small in practical magnitude.

One-way fixed effects has the lowest AIC, beating pooled OLS by about 0.750 and multi-way fixed effects by about 0.545. Pooled OLS has the lowest BIC because BIC penalizes the additional fixed-effect parameters more strongly. A responsible report names the criterion used rather than selecting whichever statistic supports the preferred model.

Fixed Effects Regression Coefficient Interpretation

G2 accounts for most of the fitted predictive relationship. Holding G1, the remaining numeric terms and all fixed-effect groups constant, a one-point increase in G2 is associated with an expected 0.8792-point increase in G3. G1 contributes an additional 0.1313 points per unit. One additional previous failure is associated with a 0.2178-point lower adjusted G3.

The four fixed-effect dummy coefficients are small relative to the unadjusted group-mean differences because prior grades and other numeric predictors explain much of the between-group variation. This is a central advantage of regression adjustment: raw group differences and conditional fixed-effect coefficients answer different questions.

Fixed-Effect Group Means and Adjusted Group Coefficients

Unadjusted Outcome Means

Why Raw Means and Fixed-Effect Coefficients Differ

The raw higher-education difference is 3.4788 G3 points, but the adjusted higher_yes coefficient is only 0.1870 and nonsignificant. Students intending higher education also tend to differ in G1, G2, failures and other predictors. The fixed-effect coefficient estimates the remaining difference after those covariates are controlled.

The same pattern occurs for school. GP students average 1.9264 points above MS students, but the adjusted MS coefficient is -0.1759. Most of the raw school difference is explained by the numeric predictor distribution, especially prior grades.

Combined Fixed-Effect Cell Means

The highest observed combined-cell mean is 13.2394 for GP, F, U, higher=yes students (n = 188). The lowest is 7.8333 for MS, F, R, higher=no students (n = 12). Several cells are small, including a four-record GP/F/R/higher=no cell. Small cells make multi-way descriptive means unstable even when the main-effect regression remains estimable.

Nested F Tests for Fixed Effects

The nested tests ask whether the added fixed-effect dummy coefficients jointly improve fit. The school-only addition has p = .1000. Adding sex, address and higher after school has p = .1479. Adding all four fixed effects to pooled OLS has p = .0896. None crosses the conventional .05 threshold.

A nonsignificant nested test does not prove that all group differences are exactly zero. It indicates that the sample does not provide sufficiently strong evidence that the larger model reduces residual variance enough to justify the added parameters under that test.

Fixed Effects Regression Assumptions

Normal residuals are not required for unbiased OLS coefficients. Normality mainly supports exact small-sample t and F inference. With 649 observations, robust standard errors and model specification are often more important than demanding a perfectly normal residual histogram.

Within, Between and Total Variation

Fixed-effects reasoning separates variation into components. Between variation concerns differences in group means. Within variation concerns deviations from each group’s mean. Total variation combines both.

In the educational combined-cell transformation, demeaned G2 and demeaned G3 correlate at 0.902094. Their simple within slope is 0.987075. The full multi-variable G2 coefficient is smaller, 0.879166, because it controls G1, studytime, failures, absences, age, parent education and the fixed-effect dummies simultaneously.

Within and between effects can differ in magnitude or even direction. A model that assumes one common coefficient may be misleading when contextual group differences and individual-level changes answer separate research questions.

Fixed Effects vs Random Effects Regression

The Hausman test is often used as one diagnostic in panel settings, but model choice should not be reduced to a single p-value. The data-generating process, substantive target, within variation and plausibility of random-effects exogeneity are essential.

For prediction, random intercepts may provide shrinkage for small groups. For estimating a within-entity association robust to stable omitted group characteristics, fixed effects may be preferable. These are different goals.

Python Chart-by-Chart Interpretation with Exact Values

Python Chart 1: G3 Outcome Distribution

G3 has N = 649, mean 11.9060, median 12, standard deviation 3.2307, first quartile 10, third quartile 14, minimum 0 and maximum 19. The most frequent score is 11 with 104 records, followed by 10 with 97, 13 with 82, 12 with 72 and 14 with 63.

The outcome is bounded and integer valued, with a dense middle and a smaller low-grade tail. Ordinary least squares treats the conditional mean as continuous. The large sample supports mean-model estimation, but residual diagnostics remain necessary because zero scores create several unusually large negative residuals.

Python Chart 2: Observed vs Fitted G3

The correlation between observed and fitted G3 is 0.923392. Fitted values range from approximately 0.0119 to 19.4504. The model’s RMSE is 1.2392 and MAE is 0.7774. Most points lie close to the 45-degree line, particularly through the central grade range.

The largest deviations occur for unusually low observed outcomes. One record has G3 = 1 but a fitted value of 10.0753, producing a residual of -9.0753. Another has G3 = 0 and fitted value 8.9991. These cases explain why strong R² can coexist with important individual prediction errors.

Python Chart 3: Residuals vs Fitted Values

The residual mean is essentially 0, the residual SD is 1.2401, and the median is -0.0139. The middle 50% ranges from -0.4803 to 0.6475. The full range is asymmetric, from -9.0753 to 5.5036.

The negative tail is driven by zero or very low G3 observations that the model predicts substantially higher. Adjusted R² does not test homoskedasticity, normality or influence. Review studentized residuals, Q-Q plots, Cook’s distance and robust standard errors before describing assumptions as satisfactory.

Python Chart 4: Adjusted R Squared Model Comparison

Adjusted R² is 0.848912 for pooled OLS, 0.849315 for one-way fixed effects and 0.849872 for multi-way fixed effects. The one-way gain over pooled OLS is only 0.000403. The multi-way gain over one-way is 0.000557, and its gain over pooled OLS is 0.000960.

The chart identifies the largest adjusted R² but not statistical significance. The corresponding nested F-test p-values are .1000, .1479 and .0896. Therefore, the chart supports a small incremental fit advantage, not strong evidence that all fixed-effect additions are necessary.

Python Chart 5: Fixed-Effect Group Means

School means are 12.5768 for GP (n = 423) and 10.6504 for MS (n = 226). Sex means are 12.2533 for F and 11.4060 for M. Address means are 11.0863 for R and 12.2633 for U. Higher-education intention means are 8.7971 for no and 12.2759 for yes.

These are unadjusted descriptive means. Their gaps are much larger than the regression dummy coefficients because G1, G2, failures and other numeric predictors differ across groups. The chart should therefore be followed by the adjusted group-coefficient chart rather than interpreted as causal group effects.

Python Chart 6: Numeric Predictor Coefficients

G2 is the dominant positive coefficient at 0.8792, 95% CI [0.8117, 0.9466]. G1 is 0.1313, CI [0.0588, 0.2037]. Failures is -0.2178, CI [-0.4060, -0.0297]. These three intervals exclude zero.

Studytime = 0.0584, absences = 0.0182, age = 0.0311, Medu = -0.0536 and Fedu = 0.0171 all have intervals crossing zero. The chart shows effect size and uncertainty; the large G2 t value of 25.6035 explains most of the model’s high fit.

Python Chart 7: Within-Transformed G2–G3 Relationship

After demeaning G2 and G3 within school × sex × address × higher cells, the correlation is 0.902094. The simple within slope is 0.987075. Demeaned G2 ranges from approximately -11.2687 to 7.1176, while demeaned G3 ranges from -11.5970 to 7.7941.

The positive within relationship shows that students with G2 above their fixed-cell mean also tend to have G3 above their fixed-cell mean. The full regression G2 coefficient is 0.8792 rather than 0.9871 because the model simultaneously controls G1 and the other numeric predictors.

Python Chart 8: Fixed-Effect Group Coefficients

The adjusted coefficients are school_MS = -0.1759, sex_M = -0.1887, address_U = 0.1077 and higher_yes = 0.1870. Their p-values are .1405, .0717, .3500 and .2978, respectively.

Every 95% interval crosses zero. The closest to conventional significance is sex_M, with interval [-0.3942, 0.0167]. The chart demonstrates how large raw group mean gaps can become small after prior academic performance and other numeric terms are controlled.

R Charts, Tables and Independent Validation

The supplied media list repeats the same URLs for charts 01 through 07 and provides a distinct -1 URL for chart 08. The R section therefore uses those supplied assets and validates the workbook results. Separate R URLs should replace repeated assets later if WordPress created distinct media files.

Open the expected Fixed Effects Regression R report PDF

R Consolidated Model-Fit Table

R Chart 1: Outcome Distribution Validation

R uses the same outcome with mean 11.9060, median 12, SD 3.2307 and range 0 to 19. The central mass between 10 and 14 and the low zero-score tail must be reproduced in any R diagnostic report.

R Chart 2: Observed vs Fitted Validation

The observed–fitted correlation is approximately 0.9234, with RMSE 1.2392 and MAE 0.7774. The R interpretation should emphasize strong aggregate fit and the same extreme low-grade overpredictions identified in Python.

R Chart 3: Residual Diagnostic Validation

The residual mean is effectively zero, but the range from -9.0753 to 5.5036 is asymmetric. The R workflow should supplement this figure with Q-Q, influence and heteroskedasticity checks rather than treating adjusted R² as an assumption test.

R Chart 4: Adjusted R Squared Comparison Validation

R reproduces the sequence .848912, .849315 and .849872. The difference between the highest and lowest value is less than .001, so model-selection discussion must include nested tests and information criteria.

R Chart 5: Group Means Validation

The R chart should reproduce the GP/MS, F/M, R/U and no/yes mean differences. The largest raw gap is higher=yes versus higher=no, 12.2759 versus 8.7971, but the adjusted higher_yes coefficient is only 0.1870.

R Chart 6: Numeric Coefficient Validation

R confirms the three intervals away from zero: G2 0.8792, G1 0.1313 and failures -0.2178. The remaining numeric predictors have confidence intervals that include zero.

R Chart 7: Within-Transformed Relationship Validation

The transformed relationship has correlation approximately 0.9021 and simple slope approximately 0.9871. It provides a visual explanation of within-group adjustment but is not a replacement for the complete multivariable coefficient table.

R Chart 8: Fixed-Effect Coefficient Validation

The R chart validates the four adjusted group terms: -0.1759 for school_MS, -0.1887 for sex_M, 0.1077 for address_U and 0.1870 for higher_yes. None has a 95% interval fully away from zero.

SPSS Fixed Effects Regression Output Interpretation

The confirmed SPSS output is available at the Cox-independent Fixed Effects Regression PDF link below. The unrelated Elastic Net SPSS PDF is not used.

Open the Fixed Effects Regression SPSS output PDF

In SPSS, fixed group effects can be represented through dummy variables in REGRESSION, categorical factors in UNIANOVA/GLM or repeated/panel structures in procedures appropriate to the design. For the workbook specification, the simplest equivalent is ordinary linear regression with four dummy variables and eight numeric predictors.

Excel Worked Fixed Effects Regression Results

The workbook contains the complete 649-row model, formula-driven fitted values and residuals, model summaries, coefficients, group summaries and nested F tests. It is not a small demonstration workbook; it directly reproduces the full multi-way model results.

Excel Model Summary

Excel Prediction Formula

For each row, the workbook creates four dummy variables: school_MS, sex_M, address_U and higher_yes. The fitted-value formula multiplies each data value by its corresponding coefficient. Residual equals observed G3 minus predicted G3. Squared residuals sum to SSE, while deviations from the outcome mean sum to SST.

Core Excel Formulas

Nested F-Test Formula

Here, q is the number of added fixed-effect parameters. The workbook reports p = .0896 for adding all four fixed effects to pooled OLS, so the expanded model is not significant at .05 under the nested F test.

Python, R, SPSS and Excel Workflows

Fixed Effects Regression Code

Python Code

import pandas as pd
import statsmodels.formula.api as smf
from statsmodels.stats.anova import anova_lm

df = pd.read_csv("dataset.csv")

numeric = [
    "G1", "G2", "studytime", "failures",
    "absences", "age", "Medu", "Fedu"
]
required = ["G3", *numeric, "school", "sex", "address", "higher"]
work = df[required].dropna().copy()

pooled_formula = "G3 ~ " + " + ".join(numeric)
one_way_formula = pooled_formula + " + C(school)"
multi_way_formula = (
    pooled_formula
    + " + C(school) + C(sex) + C(address) + C(higher)"
)

pooled = smf.ols(pooled_formula, data=work).fit()
one_way = smf.ols(one_way_formula, data=work).fit()
multi_way = smf.ols(multi_way_formula, data=work).fit()

print(multi_way.summary())
print(anova_lm(pooled, one_way))
print(anova_lm(one_way, multi_way))
print(anova_lm(pooled, multi_way))

# Heteroskedasticity-robust covariance
multi_way_hc3 = multi_way.get_robustcov_results(cov_type="HC3")
print(multi_way_hc3.summary())

R Code

df <- read.csv("dataset.csv", stringsAsFactors = TRUE)

numeric_terms <- c(
  "G1", "G2", "studytime", "failures",
  "absences", "age", "Medu", "Fedu"
)

pooled <- lm(
  G3 ~ G1 + G2 + studytime + failures +
    absences + age + Medu + Fedu,
  data = df
)

one_way <- update(pooled, . ~ . + school)

multi_way <- update(
  pooled,
  . ~ . + school + sex + address + higher
)

summary(multi_way)
anova(pooled, one_way)
anova(one_way, multi_way)
anova(pooled, multi_way)

# Robust standard errors
library(sandwich)
library(lmtest)
coeftest(multi_way, vcov = vcovHC(multi_way, type = "HC3"))

# For true panel data:
# library(fixest)
# feols(y ~ x1 + x2 | entity + year, data = panel_df)

SPSS Syntax Pattern

* Prepare reference-coded fixed-effect dummies.
COMPUTE school_MS = (school = 'MS').
COMPUTE sex_M = (sex = 'M').
COMPUTE address_U = (address = 'U').
COMPUTE higher_yes = (higher = 'yes').
EXECUTE.

REGRESSION
  /DEPENDENT G3
  /METHOD=ENTER G1 G2 studytime failures absences age Medu Fedu
  /METHOD=ENTER school_MS sex_M address_U higher_yes
  /STATISTICS COEFF OUTS R ANOVA CHANGE CI(95) COLLIN TOL
  /RESIDUALS HISTOGRAM(ZRESID) NORMPROB(ZRESID)
  /SAVE PRED RESID ZRESID COOK LEVER.

OUTPUT SAVE OUTFILE=
'D:\DATA ANALYSIS\H Regression Tests and Models\Fixed Effects Regression\SPSS_Output\spv\Fixed-Effects-Regression-SPSS-Output.spv'.

OUTPUT EXPORT
  /CONTENTS EXPORT=ALL LAYERS=PRINTSETTING MODELVIEWS=PRINTSETTING
  /PDF DOCUMENTFILE=
'D:\DATA ANALYSIS\H Regression Tests and Models\Fixed Effects Regression\SPSS_Output\pdf\Fixed-Effects-Regression-SPSS-Output.pdf'.

Excel Formula Pattern

Dummy variables:
=--(J2="MS")
=--(K2="M")
=--(L2="U")
=--(M2="yes")

Predicted fixed-effects value:
=$B$intercept
 + G1*$B$G1
 + G2*$B$G2
 + studytime*$B$studytime
 + failures*$B$failures
 + absences*$B$absences
 + age*$B$age
 + Medu*$B$Medu
 + Fedu*$B$Fedu
 + dummy_school_MS*$B$school_MS
 + dummy_sex_M*$B$sex_M
 + dummy_address_U*$B$address_U
 + dummy_higher_yes*$B$higher_yes

Robust and Clustered Standard Errors

Fixed effects change the conditional mean specification; they do not automatically correct the covariance matrix. Cross-sectional heteroskedasticity can make conventional standard errors unreliable. HC3 robust standard errors are often useful for ordinary dummy-variable regression.

In true panel data, observations from the same entity are usually correlated over time. Standard errors should generally be clustered at the entity level. When treatment or policy varies at a higher aggregation level, clustering may need to match the assignment level.

Coefficient values remain the same when only the covariance estimator changes, but standard errors, confidence intervals and p-values can change materially. Always report the covariance method.

APA-Style Reporting for Fixed Effects Regression

Full Multi-Way Model Report

Model-Comparison Report

Short Results Version

Fixed Effects Regression Reporting Checklist

Common Fixed Effects Regression Mistakes

Fixed Effects Regression Troubleshooting Guide

Dummy Variable Is Automatically Dropped

The dropped term is usually the reference category or is perfectly collinear with another variable. Inspect the design matrix and coding. One level per categorical fixed effect must be omitted when an intercept is included.

Time-Invariant Predictor Disappears in Panel FE

This is expected. Entity demeaning removes every variable that does not change within the entity. The effect cannot be separated from the entity fixed intercept without additional assumptions or a different design.

Adjusted R Squared Barely Changes

The added fixed effects may explain little residual variation after the numeric predictors are included. In this workbook, adjusted R² increases by less than .001 from pooled to multi-way fixed effects. Report the small gain accurately.

Raw Group Difference Is Large but Dummy Coefficient Is Small

The covariates explain much of the raw difference. Higher=yes students average 3.4788 points above higher=no students, but the adjusted coefficient is only 0.1870 because prior grades and other predictors differ strongly across the groups.

Python, R and SPSS Coefficients Differ

Check complete-case rows, category reference levels, contrast coding, predictor lists, transformations and covariance estimators. Dummy-label differences such as C(school)[T.MS] versus schoolMS are naming differences when coding is otherwise identical.

High R Squared but Poor Residual Tail

G1 and G2 make prediction strong overall, but several zero-score observations have residuals below -7. High R² does not eliminate outliers or influence. Review case-level diagnostics and sensitivity analyses.

Too Many Fixed Effects

High-dimensional fixed effects can consume degrees of freedom and memory. Use within/absorption algorithms such as fixest, lfe or specialized panel estimators rather than explicitly creating thousands of dummy columns.

Nested F Test Is Not Significant

Keep or remove fixed effects based on the estimand, design and confounding logic—not only one threshold. A theoretically required entity or time effect may remain in the model even when its joint incremental test is modest.

Downloads and Resources

Related Salar Cafe Guides

Final Fixed Effects Regression Conclusion

The multi-way fixed-effects model explains 85.27% of G3 variance and has adjusted R² = 0.849872. Its fitted values correlate 0.923392 with observed G3. G2 is the dominant adjusted predictor, followed by G1, while previous failures has a smaller negative effect.

The fixed-effect group dummies add only a small amount of model fit after the numeric predictors are included. Their joint improvement over pooled OLS is not significant at .05, and none of the four individual dummy intervals excludes zero. The results support strong academic prediction but limited incremental evidence for the named group intercept shifts in this specification.

The most accurate final report distinguishes raw group means from adjusted effects, states the reference categories, reports the nested tests and information criteria, checks residual and influence diagnostics, and avoids treating a cross-sectional dummy-variable model as automatically equivalent to a repeated panel fixed-effects design.