Types, Checks, Violations & Fixes
What Are Assumptions In Hypothesis Testing?
Assumptions are the basic conditions that need to be true for a statistical test to give valid results. Simply put, every hypothesis test has rules about how the data should behave. When your dataset meets these conditions, the test results are trustworthy. When it does not, the results can become biased or misleading.
These assumptions help avoid incorrect conclusions. For example, if the data is not normally distributed, running a parametric test may give inaccurate p-values or underestimate variability.
Some statistical tests rely heavily on assumptions, including:
- t-tests (require normality and independence)
- ANOVA (requires normality and equal variances)
- Regression analysis (needs linearity, homoscedasticity, independence of errors)
Types Of Assumptions Used In Hypothesis Testing
Hypothesis testing relies on two major categories of assumptions
- statistical assumptions and
- practical or research assumptions.
1. Statistical Assumptions
Statistical assumptions refer to the conditions your dataset must meet for a test to produce correct and unbiased results.
These assumptions vary depending on the test, but most parametric tests require the following:
Normality
Many hypothesis tests assume that the data (or residuals) follow a normal distribution. This is especially important for t-tests, ANOVA, and regression. Normality ensures that p-values and confidence intervals are accurate and not distorted by skewed data.
Independence
Each observation in your dataset should be independent of all others. In simple terms, one person’s score should not influence another person’s score. Violations occur in clustered data, repeated measures, or poorly designed experiments.
Homogeneity of Variance
Also known as equal variances or homoscedasticity, this assumption means that the spread of data should be similar across groups. Tests like ANOVA and independent t-tests rely heavily on this assumption. Unequal variances can distort test statistics.
Linearity
For tests like Pearson correlation and regression analysis, the relationship between variables must be linear. If the relationship is curved or non-linear, the test may underestimate or misrepresent the strength of the relationship.
Random Sampling
Your sample must be taken randomly from the population. Random sampling reduces bias and increases the generalisability of your results. Without it, hypothesis testing becomes unreliable because the sample may not reflect the population accurately.
2. Practical / Research Assumptions
Beyond statistical conditions, hypothesis testing also depends on practical research assumptions about how the data was collected and measured.
Correctly Measured Variables
The variables used in the test must be measured accurately and consistently. Poor measurement tools, incorrect scale types, or human error can lead to invalid results, no matter how strong the statistical method is.
Reliable Data Collection Methods
Data must be gathered using a valid and replicable process. Surveys, experiments, and observations should follow standard procedures to avoid bias and ensure consistency.
Appropriate Sample Size
A small sample size can make results unstable and reduce the power of the test. A sample that is too large may detect trivial differences.
Key Assumptions For Major Hypothesis Tests
Below are the major tests used in research and the assumptions that come with each.
t-Tests (One-Sample, Independent, Paired)
A t-test compares means between groups, but it only works correctly when certain conditions are met.
| Normality | The data, or the differences between paired observations, should follow a normal distribution. This matters most for small sample sizes (n < 30). |
| Independence | Each observation must be independent of others. In independent t-tests, the two groups must not influence each other. |
| Equal Variances (Independent t-test only) | Also called homogeneity of variance, both groups should have roughly equal spread. Levene’s test is commonly used to check this. |
When Violations Are Acceptable
- With large sample sizes (n > 30), t-tests are fairly robust to violations of normality.
- If variances are unequal, you can use Welch’s t-test as a valid alternative.
- For non-normal data, you can switch to a non-parametric test like the Mann, Whitney U test or Wilcoxon Signed-Rank test.
ANOVA
Analysis of Variance (ANOVA) compares means across three or more groups. Its assumptions include:
| Independence of Observations | Participants or measurements must not influence one another. This is the most crucial assumption in ANOVA. |
| Homogeneity of Variance | The variance across the groups should be similar. If this assumption is violated, you can use Welch’s ANOVA or a non-parametric alternative. |
| Normal Distribution of Residuals | Residuals (differences between observed and predicted values) should be normally distributed. ANOVA is quite robust to minor deviations, especially with larger samples. |
Chi-Square Test
The Chi-Square test is used for categorical data to test relationships between variables.
| Expected Frequencies $ge 5$ | At least 80% of the cells should have expected counts of 5 or more. Low expected values make the $chi^2$ (Chi-square) test unreliable. |
| Independent Categories | Each participant or observation must appear in one category only. No repeated measures or paired data are allowed (i.e., observations are independent). |
| Random Sampling | Data must come from a random and representative sample to ensure the test reflects the population accurately. |
Correlation (Pearson & Spearman)
Correlation tests measure the strength and direction of the relationship between two variables.
| Linearity | Pearson correlation requires a linear relationship between the two variables. If the relationship is curved, the Pearson coefficient ($r$) becomes misleading. |
| Homoscedasticity | The variability (spread) of the data points around the regression line should remain constant across the range of values for the independent variable. Unequal spread reduces the accuracy of the correlation and subsequent regression. |
| Normality (for Pearson) | Both variables should be approximately normally distributed. This is a technical assumption for inference (p-values, confidence intervals) but is not strictly required for the calculation of the Pearson $r$ itself. It is not required for Spearman correlation, which is rank-based. |
| Type of Data | Pearson requires continuous (interval or ratio) data. Spearman requires at least ordinal data, making it more flexible. |
Linear Regression
Regression predicts one variable based on another and therefore comes with several assumptions.
| Linear Relationship | The relationship between the independent variable(s) and the dependent variable must be linear. |
| Independence of Errors | Residuals (errors) must be independent of one another. The Durbin–Watson test is often used to check this assumption. |
| Normal Distribution of Errors | Residuals should follow a normal distribution. This is important for calculating valid confidence intervals and $p$-values. |
| No Multicollinearity | Independent variables should not be too highly correlated with each other. High multicollinearity can make coefficient estimates unstable. |
| Homoscedasticity | The variance of residuals should remain constant across all levels of the predictor variable(s). Unequal spread (heteroscedasticity) results in biased standard errors. |
How To Check These Assumptions
Below are simple and beginner-friendly ways to verify each assumption using commonly available tools like SPSS, R, Python, Excel, or JASP.
Normality Tests
Normality means your data follows a bell-shaped curve. Here are easy ways to check it:
Shapiro-Wilk Test
This test evaluates whether your data significantly deviates from a normal distribution.
- Recommended for small to moderate sample sizes (n < 2000).
- A p-value > .05 suggests normality.
Kolmogorov-Smirnov Test
A general test for normality, especially for larger datasets.
- Works similarly to Shapiro-Wilk.
- A p-value > .05 indicates no significant deviation from normality.
Q-Q Plots (Quantile-Quantile Plots)
A visual method where points falling along the diagonal line indicate normality.
- Easy to interpret for beginners.
- Helpful when sample sizes are large and tests become too sensitive.
Homogeneity of Variance
This assumption checks whether groups have similar variability.
Levene’s Test
The most widely used test for equal variances.
- A p-value > .05 means variances are equal.
- Works well even when data is not perfectly normal.
Bartlett’s Test
A classical test for homogeneity of variance.
- Best used when data is normally distributed.
- More sensitive to normality violations compared to Levene’s.
Independence
Independence is mostly about research design rather than calculations.
Study Design Considerations
Ask yourself:
- Were participants selected randomly?
- Did one participant’s response influence another?
- Are there repeated measures or clustered samples?
If yes, independence may be violated.
Durbin–Watson Test (for regression)
Used to check whether regression residuals are independent.
- Values close to 2 indicate independence.
- Values near 0 or 4 suggest autocorrelation.
Linearity
Linearity ensures the relationship between variables is straight-line shaped.
Scatterplots
Plot the two variables against each other.
- A roughly straight-line pattern indicates linearity.
- Curves or waves suggest non-linear relationships.
Residual Plots
Plot residuals against predicted values.
- A random cloud of points supports linearity.
- Patterns, curves, or funnels signal violations.
What Happens When Assumptions Are Violated
Ignoring assumptions can lead to serious statistical problems. Even small violations can distort results and lead to incorrect conclusions.
Biased Estimates
Coefficient estimates, means, or effect sizes may no longer reflect reality accurately.
Incorrect p-values
P-values may become too large or too small, causing researchers to accept or reject hypotheses incorrectly.
Reduced Reliability of Conclusions
Hypothesis tests lose their trustworthiness, making your findings questionable or invalid.
How To Fix Or Handle Assumption Violations
If your data does not meet the assumptions, there are practical methods to correct or work around the problem.
1. Data Transformation (Log, Square Root, Box–Cox)
Transformations can help normalise data, reduce skewness, or stabilise variances.
- Log transformation: helpful for right-skewed data
- Square root transformation: useful for count data
- Box–Cox: a flexible option for many types of skewness
Using Non-Parametric Tests
If assumptions are severely violated, switch to tests that do not assume normality. Example alternatives include:
- Mann-Whitney U instead of independent t-test
- Wilcoxon Signed-Rank instead of paired t-test
- Kruskal–Wallis instead of ANOVA
- Spearman correlation instead of Pearson correlation
Bootstrapping
A resampling technique that generates thousands of simulated samples.
- Useful when normality is violated
- Ideal for small sample sizes
- Provides more accurate confidence intervals
Robust Statistical Methods
Modern statistics offer tests that are less sensitive to assumption violations, such as:
- Welch’s t-test (unequal variances)
- Welch’s ANOVA
- Robust regression methods
Increasing Sample Size
Larger samples reduce the impact of non-normality and provide more stable estimates.
- Particularly effective when dealing with skewed distributions
- Not always practical, but very helpful when possible
Frequently Asked Questions
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