Random selection reduces sampling bias and ensures that data from your sample is actually typical of the population. Parametric tests can be used to make strong statistical inferences when data are collected using probability sampling.
While non-probability samples are more likely to be biased, they are much easier to recruit and collect data from. Non-parametric tests are more appropriate for non-probability samples, but they result in weaker inferences about the population. Keep in mind that external validity means that you can only generalize your conclusions to others who share the characteristics of your sample.
If you apply parametric tests to data from non-probability samples, be sure to elaborate on the limitations of how far your results can be generalized in your discussion section. Your participants are self-selected by their schools. Example: Sampling correlational study Your main population of interest is male college students in the US.
Using social media advertising, you recruit senior-year male college students from a smaller subpopulation: seven universities in the Boston area. Your participants volunteer for the survey, making this a non-probability sample. Calculate sufficient sample size Before recruiting participants, decide on your sample size either by looking at other studies in your field or using statistics.
There are many sample size calculators online. Different formulas are used depending on whether you have subgroups or how rigorous your study should be e. As a rule of thumb, a minimum of 30 units or more per subgroup is necessary.
See an example. By visualizing your data in tables and graphs, you can assess whether your data follow a skewed or normal distribution and whether there are any outliers or missing data. A normal distribution means that your data are symmetrically distributed around a center where most values lie, with the values tapering off at the tail ends.
In contrast, a skewed distribution is asymmetric and has more values on one end than the other. The shape of the distribution is important to keep in mind because only some descriptive statistics should be used with skewed distributions. Extreme outliers can also produce misleading statistics, so you may need a systematic approach to dealing with these values. Measures of central tendency describe where most of the values in a data set lie.
Three main measures of central tendency are often reported:. However, depending on the shape of the distribution and level of measurement, only one or two of these measures may be appropriate. For example, many demographic characteristics can only be described using the mode or proportions, while a variable like reaction time may not have a mode at all. Measures of variability tell you how spread out the values in a data set are. Four main measures of variability are often reported:.
Once again, the shape of the distribution and level of measurement should guide your choice of variability statistics. The interquartile range is the best measure for skewed distributions, while standard deviation and variance provide the best information for normal distributions. Using your table, you should check whether the units of the descriptive statistics are comparable for pretest and posttest scores. For example, are the variance levels similar across the groups? Are there any extreme values?
If there are, you may need to identify and remove extreme outliers in your data set or transform your data before performing a statistical test. From this table, we can see that the mean score increased after the meditation exercise, and the variances of the two scores are comparable. Next, we can perform a statistical test to find out if this improvement in test scores is statistically significant in the population. Example: Descriptive statistics correlational study After collecting data from students, you tabulate descriptive statistics for annual parental income and GPA.
Next, we can compute a correlation coefficient and perform a statistical test to understand the significance of the relationship between the variables in the population. Step 4: Test hypotheses or make estimates with inferential statistics A number that describes a sample is called a statistic , while a number describing a population is called a parameter. Using inferential statistics , you can make conclusions about population parameters based on sample statistics.
You can consider a sample statistic a point estimate for the population parameter when you have a representative sample e. Using data from a sample, you can test hypotheses about relationships between variables in the population. Hypothesis testing starts with the assumption that the null hypothesis is true in the population, and you use statistical tests to assess whether the null hypothesis can be rejected or not.
Statistical tests determine where your sample data would lie on an expected distribution of sample data if the null hypothesis were true.
These tests give two main outputs:. Your choice of statistical test depends on your research questions, research design, sampling method, and data characteristics.
Parametric tests make powerful inferences about the population based on sample data. But to use them, some assumptions must be met, and only some types of variables can be used. If your data violate these assumptions, you can perform appropriate data transformations or use alternative non-parametric tests instead. A regression models the extent to which changes in a predictor variable results in changes in outcome variable s.
Comparison tests usually compare the means of groups. These may be the means of different groups within a sample e. The z and t tests have subtypes based on the number and types of samples and the hypotheses:. The correlation coefficient r tells you the strength of a linear relationship between two quantitative variables.
Specifically, emphasis is given to the conclusions of the research effort and not on the way data was gathered. This fact is surely a very important factor on the reliability of the results. In the same issue, another statistical question that needs special attention is the determination of the sample size. In many published papers the authors just refer to the size of the sample without referring to the technical details of its computation. This is another serious problem that affects the credibility of the results in a survey.
These two problems are not the only that appear in surveys in social sciences but surely they are the most usual. Other issues that occur are blurred definition of the population, problems during the collection of the data that are rarely mentioned for example replacement of selected units , non-sampling errors for example non-response, over or under coverage and others.
Generally, we can say that in many cases sampling is treated with less attention than what is needed. Researchers in the field and practitioners may refer to the classical book by Kish Other useful references are Cohen , Joseph et al. The first step in any statistical analysis is the use of descriptive statistics to present the data and try to identify any kind of trends, relationships or abnormal behavior.
Analysis based on descriptive statistics or exploratory data analysis usually makes no stochastic assumptions. A first approach in parametric tests is to use the classic hypothesis tests and confidence intervals. Apart from that there are also other statistical methods that can be employed in social sciences. Regression is one of the most known methods used for analyzing relationships between variables.
The main objectives of a regression analysis is to check if there is an association between variables, to identify the strength of this relationship and to conclude to a regression equation that is used to describe this relationship.
There are several forms of regression modeling, for example, linear regression, logistic regression and regression discontinuity. There are also other aspects of the regression methodology but we confine ourselves to these cases.
All these methodologies have been extensively used in real cases of social sciences. Linear regression is the simplest of these methods since it is used to model the relationship between one dependent variable and one or more explanatory variables.
In this methodology, we try to find a function to fit the values of the explanatory variable that vary linearly with the target variable. Linear regression is particularly useful since it is able to predict the value of the dependent variable given the value of the explanatory variable or variables.
We have to stress that in this method the target variable dependent variable is continuous. In logistic regression we want to obtain a nonlinear curve to fit the data when the target variable is discrete. This methodology is particularly useful in modeling a target variable having value for example Yes 0 or No 1. More formally we can say that the target variable is binomial. Our aim is to find an equation that functionally connects the values of the explanatory variables to the values of the target variable.
The explanatory variables can be either continuous or categorical. If we transform the target variable to the logarithm of the odds of its values then the transformed target variable is linearly related to the explanatory variables. For more details about the linear and the logistic regression the interested reader can refer to Kutner et al. Regression discontinuity is used to compute the effect of an intervention. This methodology is able to give unbiased estimates of this intervention.
In a regression discontinuity design we use a rule to assign the intervention to a unit. This methodology is extensively used in education.
Specifically, a scoring rule is used after a test is given in a class to select the students that need more effort on the specific course.
Students with scores below a cutoff value are assigned to the group that will spend more time studying and students with scores above the cutoff value are assigned to the comparison group, or vice versa.
The effect of the intervention is estimated as the difference in the mean outcome of the treatment group and the comparison group. A regression line or curve is estimated for the two groups treatment and comparison groups , and the difference in the mean of these regression lines at the cutoff value of the measured variable is the estimate of the effect of the intervention.
A detailed description of regression discontinuity is given in Riley-Tilman and Burns and Jacob and Zhu Analysis of variance ANOVA is a well-known method used to compare several means at the same time using a fixed confidence level. The data used are the results of an experiment. There is a continuous dependent variable, and one or more qualitative independent variables categorical or nominal variables.
The design of the experiment must be done in such a way that it will not affect its results. For example, a completely randomized experiment does not affect the output of the experiment. However, the choice of the design of the experiment affects which analysis of variance method will be used. There are a lot of different designs of experiments and analysis of variance methods for several different cases.
Regression analysis and analysis of variance are closely related. If we use dummy variables as independent variables in analysis of variance then the analysis becomes regression analysis.
However, there is a serious difference between the two methods. In the analysis of variance if the design of the experiment is properly done, we may conclude that there is causality the independent variable has a causal effect on the dependent variable. On the other hand, in regression analysis a statistically significant effect may mean causality or not a statistically significant result does not necessarily mean causal effect.
The analysis of variance tests the independence of the response and explanatory variables. If we decide that there is this type of dependence then we have to do extra analysis to identify which means are different and to what extent.
The analysis of variance assumes that the samples in the groups categories of the independent variable are independent. This means that each group has a different sample of subjects. However, there are cases where each group has the same sample of subjects. Apparently, the samples are then dependent and of course we have to take this fact into consideration to reach credible results.
This case is called repeated measures analysis of variance. For more information on this topic, see Agresti and Finlay and Cohen and Lea Assume that a researcher wants to use the ANOVA and apart from the dependent variable and a categorical variable factor , data for one or more quantitative variables measured on each experimental unit are available.
Then, if these variables have an effect on the outcome of the experiment, they can be used in the model as independent variables. Such variables are called covariates or concomitant variables. The analysis involving all these variables is called analysis of covariance. Although the model is more complex by including the extra variables, the profit is that the error variance is reduced. Another very useful class of models is mixed models. Mixed models contain both fixed and random effects.
They are particularly useful in social sciences when we have repeated measurements. Moreover, in the case of missing data, which are very common in sample surveys, mixed models offer a strong alternative to methods like ANOVA for repeated measures.
Their drawback is that estimation is more difficult along with the fact that we end up to have a more complex model. A useful class of models is also the semiparametric models or even better the semiparametric regression models. These regression models include both parametric and nonparametric components. They are used when the usual parametric models do not have a satisfactory performance. More about nonparametric methods are given in Section 5. Another very useful method is robust regression.
Keeping in mind the usefulness of linear regression, its wide applicability and acceptance between the researchers it is natural to propose a method that overcomes the difficulty to fulfill its assumptions.
Robust regression is used to avoid the effect of outliers. One approach is to use the M-estimators and another one is to replace the normal distribution in the assumptions with a heavy-tailed distribution. Undoubtedly methods like linear regression and ANOVA have been used to an enormous extent in social sciences but many times without the proper accuracy in the details. We believe that much of the work done could be improved using the more advanced models presented in this section.
For more details the reader could refer to Christensen and Rencher and Schaalje For robust regression a useful reference is Rousseeuw and Leroy In social statistics the vast amount of research is based on parametric methods. However, many parametric methods are based on strong assumptions that are disregarded most of the times.
This has serious effect on the justification of the results. The alternative in this case is to use nonparametric statistical methods. Nonparametric statistics do not rely on a specific family of probability distributions and there is no assumption about the probability distributions of the variables used. Therefore it is an ideal collection of methods for handling real data that most of the times fail to follow these strong assumptions of parametric inference.
There is a number of techniques that are already popular among the researchers in social sciences. Such techniques are certain hypothesis tests like Wilcoxon Signed rank test, Mann-Whitney test and Kruskal-Wallis tests. Other used techniques are the Spearman correlation coefficient, the runs test and normality tests.
For a detailed review of such techniques the interested reader can refer to Corder and Foreman However, there is a number of other nonparametric methods that have been developed and are already famous among statisticians that have not gained much attention between the researchers in social sciences.
Such methods are the jackknife and the bootstrap methods. Jacknife can be used to compute the bias and the variance of an estimator whereas bootstrap estimates the variance and the distribution of a statistic or it is used to construct confidence intervals. It must be noted that both these methods are computationally demanding.
Nevertheless, they can be very useful in social sciences especially in the cases of complex estimators of parameters that need to be further studied. Another useful method is nonparametric regression. The usual linear regression is a heavily used method in social sciences. However, its assumptions are very rarely referred due to the fact that they rarely hold.
Nonparametric regression is a solution in that case. It is able to answer the initially stated problem that led to regression with flexibility in terms of the assumed model. Other interesting nonparametric methods are the ones used for density estimation like cross-validation and density estimation.
These methods estimate the probability distribution function using just the data. They can be used in cases where the distribution of the data is unknown and difficult to be computed analytically. For more details about these methods the interested reader could refer to Wasserman Usually in social sciences and generally in real problems more than one variable is involved.
These variables need to be considered together since most of the times they are related. Statistical tests work by calculating a test statistic — a number that describes how much the relationship between variables in your test differs from the null hypothesis of no relationship. It then calculates a p -value probability value. The p -value estimates how likely it is that you would see the difference described by the test statistic if the null hypothesis of no relationship were true.
If the value of the test statistic is more extreme than the statistic calculated from the null hypothesis, then you can infer a statistically significant relationship between the predictor and outcome variables. If the value of the test statistic is less extreme than the one calculated from the null hypothesis, then you can infer no statistically significant relationship between the predictor and outcome variables.
You can perform statistical tests on data that have been collected in a statistically valid manner — either through an experiment , or through observations made using probability sampling methods. For a statistical test to be valid , your sample size needs to be large enough to approximate the true distribution of the population being studied.
If your data do not meet the assumptions of normality or homogeneity of variance, you may be able to perform a nonparametric statistical test , which allows you to make comparisons without any assumptions about the data distribution. If your data do not meet the assumption of independence of observations, you may be able to use a test that accounts for structure in your data repeated-measures tests or tests that include blocking variables.
The types of variables you have usually determine what type of statistical test you can use. Quantitative variables represent amounts of things e. Types of quantitative variables include:. Categorical variables represent groupings of things e. Types of categorical variables include:. Choose the test that fits the types of predictor and outcome variables you have collected if you are doing an experiment , these are the independent and dependent variables.
Consult the tables below to see which test best matches your variables. Scribbr Plagiarism Checker. Parametric tests usually have stricter requirements than nonparametric tests, and are able to make stronger inferences from the data. They can only be conducted with data that adheres to the common assumptions of statistical tests. The most common types of parametric test include regression tests, comparison tests, and correlation tests. Regression tests look for cause-and-effect relationships.
They can be used to estimate the effect of one or more continuous variables on another variable. Comparison tests look for differences among group means. They can be used to test the effect of a categorical variable on the mean value of some other characteristic. T-tests are used when comparing the means of precisely two groups e.
Correlation tests check whether variables are related without hypothesizing a cause-and-effect relationship. These can be used to test whether two variables you want to use in for example a multiple regression test are autocorrelated.
This flowchart helps you choose among parametric tests. For nonparametric alternatives, check the table above.
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