RESEARCH GUIDE
Applying Statistical Methods in Research
Checking That the Assumptions Hold in Your Data
Statistical test assumptions are not boxes to tick after running a chosen test: each method rests on conditions that must hold well enough in the dataset at hand. Here the checks are set out in order, with what to do when a condition fails.
Statistical test assumptions are conditions of two kinds: those fixed by the design, such as independence of observations, and those that can be examined in the data, such as residual shape, unequal spread, influential cases or sparse categories. Choosing a method therefore pairs it with a set of checks, which are judged together through plots, diagnostics and robustness, not through a single significance test. Each method’s own conditions are set out under Statistical Methods. OnlineSPSS, a specialist statistical consultancy for academic, clinical and professional research, examines these conditions in the analyses it carries out, while the conclusions remain the researcher’s.
ASSUMPTION GROUPS
Kinds of Assumptions and How Each Is Checked
Assumptions fall into a few groups, and each group is checked differently, some from how the data were collected and some from the fitted model.
Independence of Observations Set by the design and not by a test: repeated, matched or clustered data break it, so the check is a question about how the data were collected.
Skew and Outliers in Residuals For models with a continuous outcome, residual plots show skew and outliers, and in moderate to large samples mild departures usually matter less than a few extreme values.
Equal Spread Unequal variances across groups or fitted values are judged from plots and, where doubtful, handled with a robust standard error or a method that does not assume equal spread.
Linearity and Form A straight-line relationship is assumed unless the model says otherwise, and residual plots against each predictor show curvature that a transformation or added term can capture.
Collinearity and Influence Highly correlated predictors inflate standard errors, and a few influential cases can shift an estimate, so both are examined before any coefficient is interpreted.
Sparse Categories Categorical tests rely on adequate expected counts in each cell, and a table with sparse cells calls for an exact method, categories merged where that makes sense or a simpler model.
CHECK COMPARISON
Diagnostic Plots, Formal Tests and Robust Methods
Each tool answers a different question about an assumption, and none of them settles every question on its own.
| Feature | Diagnostic plots | Formal tests | Robust or alternative methods |
|---|---|---|---|
| What it gives | A picture of curvature, outliers or unequal spread | A p-value for one specific departure | An estimate less sensitive to the departure |
| Main limitation | The reading is partly subjective | A pass does not show that the assumption holds | May answer a slightly different question |
| Best used for | The first look at every fitted model | Supporting a judgment when a plot is ambiguous | Showing whether the conclusion survives the departure |
| Sample size | Informative at any size, though noisy when small | Weak when small and sensitive to trivial departures when large | Some, such as robust standard errors, behave poorly in small samples, and most lose a little efficiency if the assumption held |
| Reported as | A note on what was inspected, with plots in a supplement | The statistic and p-value beside the plots | Both results, with the primary one named in advance |
KEY PLANNING DECISIONS
Decisions That Come Before the Output
Checking goes best when its rules are set before the results are seen, and each of these choices makes that possible.
What the Conclusion Rests On Identify which conditions the main conclusion depends on, since normal errors matter more for a prediction interval or a small sample than for a mean difference in a large one, while independence matters for every inference.
Inspecting the Data First Assumption checks before analysis begin with distributions, missing values and coding errors, examined with plots and summaries, since many assumption problems begin as data problems.
Observations Against Parameters Count the observations or events available against the parameters planned, because assumptions fail quietly when a model is too large for its data.
The Order of the Checks Judge design-based conditions first, then plots of the fitted model, then sensitivity to the doubtful points, so that each step informs the next.
What Counts as Too Far Decide in advance, with a plot or a threshold, how large a departure would lead to an alternative method, so the switch is not chosen after seeing which result is preferred.
A Record of Every Check Keep the diagnostics, the decisions and the code, so the checks can be repeated, described in the methods and defended if a reviewer asks.
COMMON ISSUES
How Assumption Checks Lead Researchers Astray
The checks themselves are straightforward, and what goes wrong is usually the conclusion drawn from them.
Transformation Without Interpretation A log or square-root transformation can stabilize spread, but results are then on the transformed scale, so the interpretation and any back-transformed estimate are stated.
Rank-Based Tests as a Universal Fix A rank-based test compares ordering and not means, and it does not repair dependent observations, so it is chosen for what it estimates and not only to avoid an assumption.
Linear-Model Checks on Every Model Residual normality and equal spread are checks for linear models, so a logistic or count model is judged instead by its fit, its link function and, for counts, whether the variance exceeds what the model allows.
Pretests Before Every Comparison Running a variance-equality test and choosing the main test from its p-value is a two-stage procedure with its own error rates, which Welch's approach for two groups sidesteps.
Outliers Removed to Improve Fit Dropping extreme values because they spoil a diagnostic changes the sample, so each exclusion rests on a data error or a stated rule, and results are shown with and without them.
Group Differences Hidden in Pooled Plots Residuals pooled across groups can hide a group with a different spread or shape, so plots are read by group when the aim is to compare groups.
REPORTING
Documenting the Checks in a Methods Section
A few sentences per model, naming what was inspected and what followed from it, usually give a reviewer enough to judge the checks.
Careful reporting shows in a methods section that names each model, the assumptions examined, the plots or diagnostics used, what was found and what was done where a condition looked doubtful. Full diagnostic plots can sit in a supplement, while the main text carries one sentence per model. The journal’s author guidelines or the program’s handbook set how much detail is expected, and a statement that assumptions were met, with nothing examined behind it, gives a reviewer nothing to judge.
STATISTICAL METHODS
Where the Conditions of Each Family Are Set Out
Every family has its own conditions, from equal spread in group comparisons to censoring in follow-up studies, and each category gathers the method pages that set them out.
Comparative & Hypothesis Testing
Independence, spread and ordinal outcomes
Regression & Predictive Modeling
Residuals, collinearity and influence
Multivariate, Scale & Reliability
Sampling adequacy and scale structure
Structural & Latent Variable Modeling
Indicator distributions and model fit
Longitudinal & Multilevel Analysis
Dependence from repeated and clustered data
Clinical, Survival & Epidemiological
Censoring and constant hazard ratios
ANALYSIS SERVICES
Checks Built Into the Analysis, Not Added Afterward
Diagnostics are most useful before the model is final, though a finished analysis can be examined too, and each card matches one of those moments.
Research Planning & Study Design
The method named for each question with the conditions it needs, the fallback if they fail and a sample size that fits the model's complexity.
Quantitative Data Analysis
The chosen method run on your data, with each diagnostic read against the design and a sensitivity analysis where a condition is doubtful.
Results Reporting & Interpretation
Assumption checks summarized in methods wording that a reviewer can follow, with the diagnostic plots prepared for a supplement.
Independent Statistical Review
An analysis already run is reviewed for whether its checks match the method and whether the conclusions survive the departures found.
FREQUENTLY ASKED QUESTIONS
Questions About Statistical Test Assumptions
How the conditions of common tests relate, what to do when one fails, how a small sample changes the checks and where robust methods stop are the topics of this set.
Do a t-test, ANOVA and regression have different assumptions?
Not in kind: in their classical forms all three are linear models for a continuous outcome, so they share the conditions of independent observations, a correctly specified mean, constant spread and errors that are not badly skewed or driven by extreme values. What differs is which check comes naturally, the spread within each group for a t-test or ANOVA and residuals against fitted values and predictors for a regression. Treating them as separate rule sets leads to checks that are repeated or missed.
What to do when assumptions are violated in my data?
The remedy depends on which assumption fails: a robust standard error for unequal spread, a model that fits the outcome better, a transformation whose scale is then interpreted, a rank-based method that answers a differently framed question, or a model for dependence when observations are clustered. A sensitivity analysis shows whether the conclusion depends on the choice, and any remaining limitation is stated in the report.
How should assumptions be judged when the sample is small?
In a small sample, plots are noisy and formal tests have little power, so passing a test says very little about whether a condition holds. More weight then falls on the design and on how the outcome behaved in similar studies, and methods that rely less on a distributional form, such as exact or permutation tests, are worth considering where they answer the same question. Resampling does not make a small sample more informative.
Can a robust method replace assumption checking altogether?
No. A robust method relaxes particular assumptions, such as equal spread or a symmetric error distribution, and still depends on others, above all independence of observations and a sensible model. Checking shows which assumption is the concern, and the robust method is then chosen to address that one.
How do you protect my research data and confidentiality?
Datasets and analysis output often carry sensitive material, and your data and project details are handled confidentially. Project information is not published, and researcher names and institutions are not used publicly without permission. An NDA is available where required and can be arranged before sensitive materials are shared. Read more about confidentiality and data security.
What happens when I request an assessment, and how long does it take?
You share your research question, design and relevant materials, such as a protocol, an analysis plan or a dataset, and a statistically trained Research Coordinator reviews the request and responds the same day. The analysis, deliverables, timeline and one fixed project fee are then confirmed before any work begins, so the timeline follows from the agreed scope.
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