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Statistical conclusion validity is the degree to which conclusions about the relationship among variables based on the data are correct or "reasonable". This began as being solely about whether the statistical conclusion about the relationship of the variables was correct, but now there is a movement towards moving to "reasonable" conclusions that use: quantitative, statistical, and qualitative data. [1] Fundamentally, two types of errors can occur: type I (finding a difference or correlation when none exists) and type II (finding no difference or correlation when one exists). Statistical conclusion validity concerns the qualities of the study that make these types of errors more likely. Statistical conclusion validity involves ensuring the use of adequate sampling procedures, appropriate statistical tests, and reliable measurement procedures. [2] [3] [4]
The most common threats to statistical conclusion validity are:
Power is the probability of correctly rejecting the null hypothesis when it is false (inverse of the type II error rate). Experiments with low power have a higher probability of incorrectly failing to reject the null hypothesis—that is, committing a type II error and concluding that there is no detectable effect when there is an effect (e.g., there is real covariation between the cause and effect). Low power occurs when the sample size of the study is too small given other factors (small effect sizes, large group variability, unreliable measures, etc.).
Most statistical tests (particularly inferential statistics) involve assumptions about the data that make the analysis suitable for testing a hypothesis. Violating the assumptions of statistical tests can lead to incorrect inferences about the cause–effect relationship. The robustness of a test indicates how sensitive it is to violations. Violations of assumptions may make tests more or less likely to make type I or II errors.
Each hypothesis test involves a set risk of a type I error (the alpha rate). If a researcher searches or "dredges" through their data, testing many different hypotheses to find a significant effect, they are inflating their type I error rate. The more the researcher repeatedly tests the data, the higher the chance of observing a type I error and making an incorrect inference about the existence of a relationship.
If the dependent and/or independent variable(s) are not measured reliably (i.e. with large amounts of measurement error), incorrect conclusions can be drawn.
Restriction of range, such as floor and ceiling effects or selection effects, reduce the power of the experiment, and increase the chance of a type II error. [5] This is because correlations are attenuated (weakened) by reduced variability (see, for example, the equation for the Pearson product-moment correlation coefficient which uses score variance in its estimation).
Greater heterogeneity of individuals participating in the study can also impact interpretations of results by increasing the variance of results or obscuring true relationships (see also sampling error). This obscures possible interactions between the characteristics of the units and the cause–effect relationship.
Any effect that can impact the internal validity of a research study may bias the results and impact the validity of statistical conclusions reached. These threats to internal validity include unreliability of treatment implementation (lack of standardization) or failing to control for extraneous variables.
Analysis of variance (ANOVA) is a collection of statistical models and their associated estimation procedures used to analyze the differences among means. ANOVA was developed by the statistician Ronald Fisher. ANOVA is based on the law of total variance, where the observed variance in a particular variable is partitioned into components attributable to different sources of variation. In its simplest form, ANOVA provides a statistical test of whether two or more population means are equal, and therefore generalizes the t-test beyond two means. In other words, the ANOVA is used to test the difference between two or more means.
Biostatistics is a branch of statistics that applies statistical methods to a wide range of topics in biology. It encompasses the design of biological experiments, the collection and analysis of data from those experiments and the interpretation of the results.
Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of surveys and experiments.
Statistics, like all mathematical disciplines, does not infer valid conclusions from nothing. Inferring interesting conclusions about real statistical populations almost always requires some background assumptions. Those assumptions must be made carefully, because incorrect assumptions can generate wildly inaccurate conclusions.
A statistical hypothesis test is a method of statistical inference used to decide whether the data at hand sufficiently support a particular hypothesis. More generally, hypothesis testing allows us to make probabilistic statements about population parameters. More informally, hypothesis testing is the processes of making decisions under uncertainty. Typically, hypothesis testing procedures involve a user selected tradeoff between false positives and false negatives.
Nonparametric statistics is a type of statistical analysis that makes minimal assumptions about the underlying distribution of the data being studied. Often these models are infinite-dimensional, rather than finite dimensional, as is parametric statistics. Nonparametric statistics can be used for descriptive statistics or statistical inference. Nonparametric tests are often used when the assumptions of parametric tests are evidently violated.
In scientific research, the null hypothesis is the claim that the effect being studied does not exist. Note that the term "effect" here is not meant to imply a causative relationship.
A chi-squared test is a statistical hypothesis test used in the analysis of contingency tables when the sample sizes are large. In simpler terms, this test is primarily used to examine whether two categorical variables are independent in influencing the test statistic. The test is valid when the test statistic is chi-squared distributed under the null hypothesis, specifically Pearson's chi-squared test and variants thereof. Pearson's chi-squared test is used to determine whether there is a statistically significant difference between the expected frequencies and the observed frequencies in one or more categories of a contingency table. For contingency tables with smaller sample sizes, a Fisher's exact test is used instead.
In statistics, the power of a binary hypothesis test is the probability that the test correctly rejects the null hypothesis when a specific alternative hypothesis is true. It is commonly denoted by , and represents the chances of a true positive detection conditional on the actual existence of an effect to detect. Statistical power ranges from 0 to 1, and as the power of a test increases, the probability of making a type II error by wrongly failing to reject the null hypothesis decreases.
An F-test is any statistical test used to compare the variances of two samples or the ratio of variances between multiple samples. The test statistic, random variable F, is used to determine if the tested data has an F-distribution under the true null hypothesis, and true customary assumptions about the error term (ε). It is most often used when comparing statistical models that have been fitted to a data set, in order to identify the model that best fits the population from which the data were sampled. Exact "F-tests" mainly arise when the models have been fitted to the data using least squares. The name was coined by George W. Snedecor, in honour of Ronald Fisher. Fisher initially developed the statistic as the variance ratio in the 1920s.
In statistics, a spurious relationship or spurious correlation is a mathematical relationship in which two or more events or variables are associated but not causally related, due to either coincidence or the presence of a certain third, unseen factor.
A t-test is a statistical hypothesis test used to test whether the difference between the response of two groups is statistically significant or not. It is any statistical hypothesis test in which the test statistic follows a Student's t-distribution under the null hypothesis. It is most commonly applied when the test statistic would follow a normal distribution if the value of a scaling term in the test statistic were known. When the scaling term is estimated based on the data, the test statistic—under certain conditions—follows a Student's t distribution. The t-test's most common application is to test whether the means of two populations are different. In many cases, a Z-test will yield very similar results to a t-test since the latter converges to the former as the size of the dataset increases.
In statistical hypothesis testing, the alternative hypothesis is one of the proposed proposition in the hypothesis test. In general the goal of hypothesis test is to demonstrate that in the given condition, there is sufficient evidence supporting the credibility of alternative hypothesis instead of the exclusive proposition in the test. It is usually consistent with the research hypothesis because it is constructed from literature review, previous studies, etc. However, the research hypothesis is sometimes consistent with the null hypothesis.
Data dredging is the misuse of data analysis to find patterns in data that can be presented as statistically significant, thus dramatically increasing and understating the risk of false positives. This is done by performing many statistical tests on the data and only reporting those that come back with significant results.
This glossary of statistics and probability is a list of definitions of terms and concepts used in the mathematical sciences of statistics and probability, their sub-disciplines, and related fields. For additional related terms, see Glossary of mathematics and Glossary of experimental design.
In statistics, resampling is the creation of new samples based on one observed sample. Resampling methods are:
Phylogenetic autocorrelation also known as Galton's problem, after Sir Francis Galton who described it, is the problem of drawing inferences from cross-cultural data, due to the statistical phenomenon now called autocorrelation. The problem is now recognized as a general one that applies to all nonexperimental studies and to experimental design as well. It is most simply described as the problem of external dependencies in making statistical estimates when the elements sampled are not statistically independent. Asking two people in the same household whether they watch TV, for example, does not give you statistically independent answers. The sample size, n, for independent observations in this case is one, not two. Once proper adjustments are made that deal with external dependencies, then the axioms of probability theory concerning statistical independence will apply. These axioms are important for deriving measures of variance, for example, or tests of statistical significance.
Mauchly's sphericity test or Mauchly's W is a statistical test used to validate a repeated measures analysis of variance (ANOVA). It was developed in 1940 by John Mauchly.
In statistics, one purpose for the analysis of variance (ANOVA) is to analyze differences in means between groups. The test statistic, F, assumes independence of observations, homogeneous variances, and population normality. ANOVA on ranks is a statistic designed for situations when the normality assumption has been violated.
In statistics, a sequence of random variables is homoscedastic if all its random variables have the same finite variance; this is also known as homogeneity of variance. The complementary notion is called heteroscedasticity, also known as heterogeneity of variance. The spellings homoskedasticity and heteroskedasticity are also frequently used. Assuming a variable is homoscedastic when in reality it is heteroscedastic results in unbiased but inefficient point estimates and in biased estimates of standard errors, and may result in overestimating the goodness of fit as measured by the Pearson coefficient.