Factorial designs are used when there are more than one independent variables. There are several factorial designs:Independent factorial design (between groups)There are several independent variables measured using different entities.Repeated-measured (related) factorial designThere are several independent variables using the same entities in all conditions.Mixed designThere are several independent variables. Some conditions use the same entities and some conditions use different entities.INDEPENDENT FACTORIAL DESIGNS AND THE LINEAR MODELThe calculation of factorial designs is similar to that of ANOVA, but the explained variance (between-groups variance) consists of more than one independent variable. The model sum of squares (between-groups variance) consists of the variance due to the first variable, the variance due to the second variable and the variance due to the interaction between the first and the second variable. It uses the following formula:This is the model sum of squares and shows you how much variance the independent variables explain. It can be useful to see how much of the total variance each independent variable explains. This can be done by using the same formula, but then only for one independent variable. In order to achieve this, the independent variable has to be grouped together in one group (this normally increases the n, as more multiple groups are being put together in one big group). The residual sum of squares, the error variance (SSR) shows how much variance cannot be explained by the independent variables. It uses the following formula:It is the variance of a group times the number of participants in the group minus one for each group added together. The degrees of freedom are added up together too. In a two-way design, the F-statistic is computed for the two main effects and the interaction. OUTPUT FROM FACTORIAL DESIGNSA main effect should not be interpreted in the presence of a significant...
Discovering statistics using IBM SPSS statistics by Andy Field, fifth edition – Summary chapter 12
The overall fit of a linear model is tested using the F-statistic. The F-statistic is used to test whether groups are significantly different and then specific model parameters (the bs) are used to show which groups are different.
The F-statistic gives an associated p-value as well. A p-value which is smaller than 0.05 (or any set alpha) stands for a significant difference between the group means. The downside of the F-test is that it does not tell us which groups are different. Associated t-tests can show which groups are significantly different.
The null hypothesis if the F-statistic is that the group means are equal and the alternative hypothesis is that the group means are not equal. If the null hypothesis is true, then the b-coefficients should be zero. The F-statistic can also be described as the ratio of explained to the unexplained variation.
The total sum of squares is the total amount of variation within the data. This can be calculated by using the following formula:
It is the difference between each observed data point and the grand mean squared. The grand variance is the total sum of squares of all observations. It is the variation between all scores, regardless of the group from which the scores come.
The model sum of squares is calculated by taking the difference between the values predicted by the model and the grand mean. It tells us how much of the variation can be explained using the model. It uses the following formula:
It is the difference of the group mean and the grand mean squared. This value is multiplied with the number of participants in this group and these values for each group are added together.
The residual sum of squares tells us how much of the variation cannot be explained by the model. It is calculated by looking at the difference between the score obtained by a person and the mean of the group to which the person belongs. It uses the following formula:
It is the squared difference between the participant’s score (xig) and the group mean and this is done for all the participants in all the groups. The residual sum of squares can also be denoted in the following way:
One other way of denoting the residual sum of squares is the following formula:
It is the variance of a group multiplied by one less than the number of people in that group and this value is added together for all the groups. The average sum of squares (mean squares) is calculated by dividing the model sum of squares with the degrees of freedom (N-k).
ASSUMPTIONS WHEN COMPARING MEANS
There are several assumptions when
Discovering statistics using IBM SPSS statistics by Andy Field, fifth edition – Summary chapter 13Covariates are characteristics of the participants in an experiment. These are characteristics outside of the actual treatment. If a researcher wants to compare means of multiple groups using the additional predictors, the covariates, then the ANCOVA is used. Examples of covariates could be love for puppies, softness of puppy fur.
Covariates can be included in an ANOVA for two reasons:
If there are covariates, the b-values represent the differences between the means of each group and the control adjusted for the covariate.
ASSUMPTIONS AND ISSUES WITH ANCOVA
There are two new assumptions for ANCOVA that are not present with ANOVA. These assumptions are independence of the covariate and treatment effect and homogeneity of regression slopes.
The ideal case is that the covariate is independent from the treatment effect. If the covariate is not independent from the treatment effect, then the covariate will reduce the experimental effect because it explains some of the variance that would otherwise be applicable to the experiment. The ANCOVA does not control for or balance out the differences caused by the covariate. The problem of covariates potentially explaining a bit of the data and wanting to filter these confounds is using randomizing participants to experimental groups or matching experimental groups on a covariate.
Another assumption of the ANCOVA is that the relationship between covariate and outcome variable holds true for all groups of participants and not only for a few groups of participants (e.g. for both males and females and not only males). This assumption can be checked by checking the regression line for all the covariates and all the conditions. The lines should be similar.
In order to test the assumption of homogeneity of regression slopes, the ANCOVA model should be customized on SPSS to look at the independent variable x the covariate interaction.
CALCULATING THE EFFECT SIZE
The partial eta squared is the effect size which takes the covariates into account. It uses the proportion of variance that a variable explains that is not explained by other variables in the analysis. It uses the following formula:
Discovering statistics using IBM SPSS statistics by Andy Field, fifth edition – Summary chapter 14Factorial designs are used when there are more than one independent variables. There are several factorial designs:
INDEPENDENT FACTORIAL DESIGNS AND THE LINEAR MODEL
The calculation of factorial designs is similar to that of ANOVA, but the explained variance (between-groups variance) consists of more than one independent variable. The model sum of squares (between-groups variance) consists of the variance due to the first variable, the variance due to the second variable and the variance due to the interaction between the first and the second variable.
It uses the following formula:
This is the model sum of squares and shows you how much variance the independent variables explain. It can be useful to see how much of the total variance each independent variable explains. This can be done by using the same formula, but then only for one independent variable. In order to achieve this, the independent variable has to be grouped together in one group (this normally increases the n, as more multiple groups are being put together in one big group).
The residual sum of squares, the error variance (SSR) shows how much variance cannot be explained by the independent variables. It uses the following formula:
It is the variance of a group times the number of participants in the group minus one for each group added together. The degrees of freedom are added up together too. In a two-way design, the F-statistic is computed for the two main effects and the interaction.
OUTPUT FROM FACTORIAL DESIGNS
A main effect should not be interpreted in the presence of a significant interaction involving that main effect. In other words, main effects don’t need to be interpreted if an interaction effect involving that variable is significant.
Simple effects analysis looks at the effect of one independent variable at individual levels of the other independent variable. When judging interaction graphs, there are two general rules:
Discovering statistics using IBM SPSS statistics by Andy Field, fifth edition – Summary chapter 15Repeated-measures refers to when the same entities (e.g. people) participate in all conditions of an experiment or provide data at multiple points of time.
One of the assumptions of the standard linear model is that residuals are independent, which is not true for repeated-measures designs. The residuals are affected by both between-participant factors and within-participant factors. There are two solutions to this:
One of these assumptions is sphericity (circularity). This assumption states that the relationship between scores in pairs of treatment conditions is similar (e.g. the level of dependence between means is roughly equal). It states that variation within conditions are similar and that no two conditions are any more dependent than any other two. Local sphericity refers to when some conditions do have equal variance and some do not. Sphericity is not relevant if there are only two groups. It becomes relevant when there are at least three conditions.
The assumption of sphericity can be tested using Mauchly’s test. The degree of sphericity can be estimated using the Greenhouse-Geisser estimate or the Huyn-Feldt estimate. If the assumption of sphericity is not met, then there is a loss of power and the F-statistic doesn’t have the distribution it is supposed to have. In order to do post-hoc tests when you worry about whether the assumption of sphericity is violated, Bonferonni method can be used, if it is not violated, Tukey’s test can be used.
If the assumption of sphericity is violated, the degrees of freedom has to be adjusted. The degrees of freedom is multiplied by the estimate of sphericity to calculate the adjusted degrees of freedom.
F-STATISTIC OF REPEATED MEASURES DESIGN
In repeated measured designs, the within-groups variance consists of within-participant variance, as there is only one group of participants. This consists of the effect of the experiment and the error (variance not explained by the experiment). The between-groups variance now consists of the between-participant variance.
The formula for the within-entity (groups) variance is the following:
The n represents the number of scores within the person (e.g. number of experimental conditions). The total amount of variance that is explained by the experimental manipulation can be calculated by comparing the condition mean to the grand mean for all the conditions. It uses the following formula:
The total error variance (residual sum of squares), the amount of variance that cannot be explained by the experimental manipulation can be calculated in the following way:
In order to calculate the F-statistic, the mean squares have to be calculated and this can be done by dividing both the SSR and the SSM by the degrees of freedom:
Discovering statistics using IBM SPSS statistics by Andy Field, fifth edition – Summary chapter 16Mixed designs are a combination of repeated-measures and independent designs. It includes some independent variables that were measured using different entities and some independent variables that used repeated measures.
The most important assumptions of the mixed designs ANOVA are sphericity and homogeneity of variance.
Discovering statistics using IBM SPSS statistics by Andy Field, fifth edition – Summary chapter 17A multivariate analysis is used when there is more than one dependent (outcome) variable. It is possible to use several F-tests when there are several dependent variables, but this inflates the type-I error rate. A MANOVA can detect whether groups differ along a combination of dimensions. MANOVA has a greater potential power to detect an effect.
A matrix is a grid of numbers arranged in columns and rows. The values within a matric are called components or elements and the rows and columns are vectors. A square matrix has an equal number of columns and rows. An identity matrix is a matrix where the diagonal numbers are ‘1’ and the non-diagonal numbers are ‘0’. The sum of squares and cross-products (SSCP) matrices are a way of operationalize multivariate versions of the sums of squares. The matrix that represents the systematic variance (model sum of squares) is denoted by the letter ‘H’ and is called the hypothesis sum of squares and cross-products matrix (hypothesis SCCP). The matrix that represents the unsystematic variance (residual sum of squares) is denoted by the letter ‘E’ and is called the error sums of squares and cross-products matrix (error SSCP). The matrix that represents the total sums of squares for each outcome (total SSCP) is denoted by the letter ‘T’.
The cross-product is the total combined error between two variables.
THEORY BEHIND MANOVA
The total sum of squares is calculated by calculating the difference between each of the scores and the mean of those scores, then squaring those differences and adding them together.
The degrees of freedom is N-1. The model sum of squares is calculated by taking the difference between each group mean and the grand mean, squaring it, multiplying by the number of scores in the group and then adding it all together.
The degrees of freedom is the sample size of each group minus one multiplied by the number of groups. The SST and the SSM then have to be divided by their own degrees of freedom, before being divided by each other to get to the F-statistic.
The cross-product is the difference between the scores and the mean for one variable multiplied by the difference between the scores and the mean for another variable. It is similar to covariance. It uses the following formula:
For each outcome (dependent) variable, the score is taken and subtracted from the grand mean for that variable. This gives x values per participant, with x being the number of outcome variables.
The model cross-product, how the relationship between the outcome variables is influenced by the experimental manipulation, uses the following formula:
The residual cross-product, how the relationship between the outcome variables is influenced by individual differences and unmeasured variables, can be calculated
.....read more Dienes (2008). Understanding psychology as a science.” – Article summary A falsifier of a theory is any potential observation statement that would contradict the theory. There are different degrees of falsifiability, as some theories require fewer data points to be falsified than others. In other words, simple theories should be preferred as these theories require fewer data points to be falsified. The greater the universality a theory, the more falsifiable it is.
A computational model is a computer simulation of a subject. It has free parameters, numbers that have to be set (e.g. number of neurons used in a computational model of neurons). When using computational models, more than one model will be able to fit the actual data. However, the most falsifiable model that has not been falsified by the data (fits the data) should be used.
A theory should only be revised or changed to make it more falsifiable. Making it less falsifiable is ad hoc. Any revision or amendment to the theory should also be falsifiable. Falsifia
Standard statistics are useful in determining probabilities based on the objective probabilities, the long-run relative frequency. This does not, however, give the probability of a hypothesis being correct.
Subjective probability refers to the subjective degree of conviction in a hypothesis. The subjective probability is based on a person’s state of mind. Subjective probabilities need to follow the axioms of probability.
Bayes’ theorem is a method of getting from one conditional probability (e.g. P(A|B)) to the inverse. The subjective probability of a hypothesis is called the prior. The posterior is how probable the hypothesis is to you after data collection. The probability of obtaining the data given the hypothesis is called the likelihood (e.g. P(D|H). The posterior is proportional to the likelihood times the prior. Bayesian statistics is updating the personal conviction in light of new data.
The likelihood principle states that all the information relevant to inference contained in data is provided by the likelihood. A hypothesis having the highest likelihood does not mean that it has the highest probability. A hypothesis having the highest likelihood means that the data support the hypothesis the most. The posterior probability is not reliant on the likelihood.
The probability distribution of a continuous variable is called the probability density distribution. It has this name, as a continuous variable has infinite possibilities and probabilities in this distribution gives the probability of any interval.
A likelihood could be a probability or a probability density and it can also be proportional to a probability or a probability density. Likelihoods provide a continuous graded measure of support for different hypotheses.
In Bayesian statistics (likelihood analysis), the data is fixed but the hypothesis can vary. In significance testing, the hypothesis is fixed (null hypothesis) but the data can vary. The height of the curve of the distribution for each hypothesis is relevant in calculating the likelihood. In significance testing, the tail area of
.....read more “Meltzoff & Cooper (2018). Critical thinking about research: Psychology and related fields.” - Article summaryPeople can get more polarized opinions after they have read the same article if they started off with opposing beliefs. This can be due to misinterpreting the evidence (1), different motivation (2) or other cognitive malfunctions (3).
Psychologists need to be critical thinkers, because they want to know the field and want to keep up with what is going on in the field of psychology.
Critical thinking requires to check for scientific soundness and not whether a paper matches with pre-existing beliefs, assumptions or other forms of bias. Scepticism of scientists towards their own work is valuable, as they then do not show the confirmation bias. There are several expectations a critical reader should have:
“Simmons, Nelson, & Simonsohn (2011). False-positive psychology: Undisclosed flexibility in data collection and analysis allows presenting anything as significant.” - Article summary A false positive refers to an incorrect rejection of the null hypothesis. The decisions a researcher can make during the research process is called the researcher degrees of freedom. Four common degrees of freedom are choosing sample size (1), using covariates (2), choosing among dependent variables (3) and reporting subsets of experimental conditions (4). The researcher degrees of freedom can significantly increase the false positive rate.
There are six guidelines for authors to prevent the increased rate of false positives:
There are four guidelines for reviewers to prevent the increased rate of false positives:
“Nosek, Spies, & Motyl (2012). Scientific utopia: II. Restructuring incentives and practices to promote truth over publishability.” - Article summary There is a publication bias in research. Mostly positive results – that is significant – are being published and non-significant results are often not published. This leads to researchers only trying to find significant results instead of genuine effects or non-effects.
Researchers are more likely to find significant results because of flexible analysis options (1), the confirmation bias (2) and the drive to find significant results (3). It is possible to find significant results everywhere if you search hard enough. Direct replications do not occur often in research and are almost never published.
There are several practices that can increase publishability but can decrease the validity of the results:
Conceptual replication involves deliberately changing the operationalization of the key elements of the design such as the independent variable, dependent variable or both. Demonstrating the same effects with multiple operationalizations provides confidence in its conceptual interpretation. Conceptual replication is not an effective replacement for direct replication.
The peer review process offers a way to detect false results. Not publishing articles without replications would also be effective in lessening the publication bias and the number of false results in science. This could also be ineffective, because it could reduce the number of innovative research as scientists might ‘play it on safe’.
Paradigm-driven research can be used to both confirm and disconfirm prior results. It accumulates knowledge by systematically altering a procedure to investigate a theory or a research question. This includes both replication and extension of the research. One pitfall of paradigm-driven research is that the research could become about the methodology, instead of the theory. Conceptual replication prevents this.
Check-lists could provide effective when conducting research as this prevents information from being left out. A metric that determines what is worth replicating could also prove to be effective. Resource constraints to replicate some findings could be overcome by using crowdfunding. Peer reviewers judge whether a finding is important enough to be published, but they are not always capable of making that judgement.
Another solution to the publication bias and the number of false results in science is to shift the attention from publishing – make publishing trivial – to the evaluation of the research.
The solution to
.....read more Discovering statistics using IBM SPSS statistics by Andy Field, fifth edition – Summary chapter 1The research process generally starts with an observation. After the observation, relevant theories are consulted and hypotheses are generated, from which predictions are made. After that, data is collected to test the predictions and finally the data is analysed. The data analysis either supports or does not support the hypothesis. A theory is an explanation or set of principles that is well substantiated by repeated testing and explains a broad phenomenon. A theory should be able to explain all of the data. A hypothesis is a proposed explanation for a fairly narrow phenomenon or set of observations. Hypotheses are theory-driven. Predictions are often used to move from the conceptual domain to the observable domain to be able to collect evidence. Falsification is the act of disproving a hypothesis or theory. A scientific theory should be falsifiable and explain as much of the data as possible.
DATA
Variables are things that can vary. An independent variable is a variable thought to be the cause of some effect and is usually manipulated, in research. A dependent variable is a variable thought to be affected by changes in an independent variable. The predictor variable is a variable thought to predict an outcome variable (independent variable). The outcome variable is a variable thought to change as a function of changes in a predictor a predictor variable (dependent variable). The difference between dependent variables and outcome variables is that one is about experimental research and the other is applicable to both experimental and correlational research.
The level of measurement is the relationship between what is being measured and the numbers that represent what is being measured. A categorical variable is made up of categories. There are three types of categorical variables:
Nominal data can be used when considering frequencies. Ordinal data does not tell us anything about the difference between points on a scale. A continuous variable is a variable that gives us a score for each person and can take on any value. An interval variable is a continuous variable with equal differences between the intervals (e.g. the difference between a ‘9’ and a ‘10’ on a grade). Ratio variables are continuous variables in which the ratio has meaning (e.g. a rating of ‘4’ is twice as good as a rating of ‘2’). Ratio variables require a meaningful zero point. A discrete variable is a variable that can take on only certain values.
Measurement error is the discrepancy between the numbers we use to represent the thing we’re measuring and the actual value of this thing. Self-report will produce larger measurement error. Validity is whether an instrument measures what it sets out to measure. Reliability is whether an instrument
.....read more Discovering statistics using IBM SPSS statistics by Andy Field, fifth edition – Summary chapter 2Many statistical models try to predict an outcome from one or more predictor variables. Statistics includes five things: Standard error (S), parameters (P), interval estimates (I), null hypothesis significance testing (N) and estimation (E), together making SPINE. Statistics often uses linear models, as this simplifies reality in an understandable way.
All statistical models are based on the same thing:
The data we observe can be predicted from the model we choose to fit plus some amount of error. The model will vary depending on the study design. The bigger a sample is, the more likely it is to reflect the whole population.
PARAMETER
A parameter is a value of the population. A statistic is a value of the sample. A parameter can be denoted by ‘b’. The outcome of a model uses the following formula:
In this formula, ‘b’ denotes the parameter and ‘X’ denotes the predictor variable. This formula calculates the outcome from two predictors. Degrees of freedom relates to the number of observations that are free to vary. One parameter is hold constant and the degrees of freedom must be one fewer than the number of scores used to calculate that parameter because the last number is not free to vary.
ESTIMATION
The method of least squares is minimizing the sum of squared errors. The smaller the error, the better your estimate is. When estimating a parameter, we try to minimize the error in order to have a better estimate.
STANDARD ERROR
The standard deviation tells us how well the mean represents the sample data. The difference in means across samples is called the sampling variation. Samples vary because they include different members of a population. A sampling distribution is the frequency distribution of sample means from the same population. The mean of the sampling distribution is equal to the population mean. A standard deviation of the sampling distribution tells us how the data is spread around the population mean, meaning that the standard deviation of the sampling distribution is approximately equal to the standard deviation of the population. The standard error of the mean (SE) can be calculated by taking the difference between each sample mean and the overall mean, squaring these differences, adding them up and dividing them by the number of samples and taking the square root of it. It uses the following formula:
The central limit theorem states that when samples get large (>30), the sampling distribution has a normal distribution with a mean equal to the population mean and the following standard deviation:
If the sample is small (<30), the sampling distribution has a t-distribution shape.
INTERVAL ESTIMATES
It is not possible to know the parameters, thus confidence intervals are used. Confidence intervals
Discovering statistics using IBM SPSS statistics by Andy Field, fifth edition – Summary chapter 3There are three main misconceptions of statistical significance:
The use of NHST encourages ‘all-or-nothing’ thinking. A result is either significant or not. If a confidence interval contains zero, it could be that the population effect might be zero.
An empirical probability is the proportion of events that have the outcome in which you’re interested in an indefinitely large collective of events. The p-value is the probability of getting a test statistic at least as large as the one observed relative to all possible values of the null hypothesis from an infinite number of identical replications of the experiment. It is the frequency of the observed test statistic relative to all possible values that could be observed in the collective of identical experiments. The p-value is affected by the intention of the researcher as the p-values are relative to all possible values in identical experiments and sample size and time of collection of data (the intentions) could influence the p-values.
In journals, based on NHST, there is a publication bias. Significant results are more likely to get published. Researcher degrees of freedom are ways in which the researcher could influence the p-value. This could be used to make it more likely to find a significant result (e.g. by excluding some cases to make the result significant). Researcher degrees of freedom could include not using some observations and not publishing key findings.
P-hacking refers to selective reporting of significant p-values by trying multiple analyses and reporting only the significant ones. HARKing refers to making a hypothesis after data collection and presenting it as if it was made before data collection. P-hacking and HARKing makes results difficult to replicate. Tests of excess success (e.g. looking at multiple studies studying the same and calculating the probability of them all having success) are used to see whether it is likely that p-hacking or something else may have occurred.
EMBERS
There is an abbreviation for how to tackle the problems of NHST: Effect sizes (E), Meta-analysis (M), Bayesian Estimation (BE), Registration (R) and Sense (S), together making EMBERS.
SENSE
There are six principles for when using NHST in order to use your sense:
Discovering statistics using IBM SPSS statistics by Andy Field, fifth edition – Summary chapter 5A good graph has the following properties:
There are also some graph building guidelines:
HISTOGRAMS
There are different types of histograms:
BOXPLOTS
A box-plot or box-whisker diagram uses the median as the centre of the plot. It is surrounded by the quartiles which show 25% and 75% of the data. There are several types of boxplots:
BAR CHARTS
Bar charts are often used to display means. There are different types of bars:
LINE CHARTS
Line charts are bar charts but with lines instead of bars. There are two types of line charts:
SCATTERPLOTS
A scatterplot is a graph that plots each person’s score on one variable against their score on another. There are several types of scatterplots:
Discovering statistics using IBM SPSS statistics by Andy Field, fifth edition – Summary chapter 6Bias can be detrimental for the parameter estimates (1), standard errors and confidence intervals (2) and the test statistics and p-values (3). Outliers and violations of assumptions are forms of bias.
An outlier is a score very different from the rest of the data. They bias parameter estimates and have an impact on the error associated with that estimate. Outliers have a strong effect on the sum of squared errors and this biases the standard deviation.
There are several assumptions of the linear model:
The assumption of normality is mainly relevant in small samples. Outliers can be spotted using graphs (e.g. histograms or boxplots). Z-scores can also be used to find outliers.
The P-P plot can be used to look for normality of a distribution. It is the expected z-score of a score against the actual z-score. If the expected z-scores overlap with the actual z-scores, the data will be normally distributed. The Q-Q plot is like the P-P plot but it plots the quantiles of the data instead of every individual score.
Kurtosis and skewness are two measures of the shape of the distribution. Positive values of skewness indicate a lot of scores on the left side of th distribution. Negative values of skewness indicate a lot of scores on the right side of the distribution. The further the value is from zero, the more likely it is that the data is not normally distributed.
Normality can be checked by looking at the z-scores of the skewness and kurtosis. It uses the following formula:
Levene’s test is a one-way ANOVA on the deviation scores. The homogeneity of variance can be tested using Levene’s test or by evaluating a plot of the standardized predicted values against the standardized residuals.
REDUCING BIAS
There are four ways of correcting problems with the data:
Discovering statistics using IBM SPSS statistics by Andy Field, fifth edition – Summary chapter 7Non-parametric tests can be used when the assumptions of the regular statistical tests have been violated. Non-parametric tests use fewer assumptions and are robust. A non-parametric test has less power than parametric tests if the sampling distribution is normally distributed.
Ranking the data refers to giving the lowest score the rank of 1, the next highest score a rank of 2 and so on. This eliminates the effect of outliers. It does neglect the difference in magnitude between the scores. If there are two scores that are the same, there are tied ranks. These scores are ranked by the value of the average potential rank for those scores (e.g. rank 3 and 4 will become rank 3.5).
There are several alternatives to the four most used non-parametric tests:
The effect size for both the Wilcoxon rank-sum test and the Mann-Whitney test can be calculated using the following formula:
denotes the total sample size.
WILCOXON RANK-SUM TEST
This test can be used to
Discovering statistics using IBM SPSS statistics by Andy Field, fifth edition – Summary chapter 8Variance of a single variable represents the average amount that the data vary from the mean. The cross-product deviation multiplies the deviation for one variable by the corresponding deviation for the second variable. The average value of the cross-product deviation is the covariance. This is an averaged sum of combined deviation. It uses the following formula:
A positive covariance indicates that if one variable deviates from the mean, the other variable deviates in the same direction. A negative covariance indicates that if one variable deviates from the mean, the other variable deviates in the opposite direction.
Covariance is not standardized and depends on the scale of measurement. The standardized covariance is the correlation coefficient and is calculated using the following formula:
A correlation coefficient of values 
0.1 represents a small effect. Values of 0.3 represent a medium effect and values of 0.5 represent a large effect.
In order to test the null hypothesis of the correlation, namely that the correlation is zero, z-scores can be used. In order to use the z-scores, the distribution must be normal, but the r-sampling distribution is not normal. The following formula adjusts r in order to make the sampling distribution normal:
The standard error uses the following formula:
This leads to the following formula for z:
The null hypothesis of correlations can also be tested using the t-score with degrees of freedom N-2:
The confidence intervals for the correlation uses the same formula as all the other confidence intervals. These values have to be converted back to a correlation efficient using the following formula:
CORRELATION
Normality in correlation is only important if the sample size is small (1), there is significance testing (2) or there is a confidence interval (3). The assumptions of correlation are normality (1) and linearity (2).
The correlation coefficient squared (R2
) is a measure of the amount of variability in one variable that is shared by the other. Spearman’s correlation coefficient (rs
) is a non-parametric statistic that is sued to minimize the effects of extreme scores or the effects of violations of the assumptions. Spearman’s correlation coefficient works best if the data is ranked. Kendall’s tau, denoted by τ, is a non-parametric statistic that is used when the data set is small with a large set of tied ranks.
A biserial or point-biserial correlation is used when a relationship between two variables is investigated when one of the two variables is dichotomous (e.g. yes
.....read more Discovering statistics using IBM SPSS statistics by Andy Field, fifth edition – Summary chapter 9Any straight line can be defined by the slope (1) and the point at which the line crosses the vertical axis of the graph (intercept) (2). The general formula for the linear model is the following:
Regression analysis refers to fitting a linear model to data and using it to predict values of an outcome variable (dependent variable) from one or more predictor variables (independent variables). The residuals are the differences between what the model predicts and the actual outcome. The residual sum of squares is used to assess the ‘goodness-of-fit’ of the model on the data. The smaller the residual sum of squares, the better the fit.
Ordinary least squares regression refers to defining the regression models for which the sum of squared errors is the minimum it can be given the data. The sum of squared differences is the total sum of squares and represents how good the mean is as a model of the observed outcome scores. The model sum of squares represents how well the model can predict the data. The larger the model sum of squares, the better the model can predict the data. The residual sum of squares uses the differences between the observed data and the model and shows how much of the data the model cannot predict.
The proportion of improvement due to the model compared to using the mean as a predictor can be calculated using the following formula:
This value represents the amount of variance in the outcome explained by the model relative to how much variation there was to explain. The F-statistic can be calculated using the following formulas:
‘k’ represents the degrees of freedom and denotes the number of predictors.
The F-statistic can also be used t test the significance of with the null hypothesis being that is zero. It uses the following formula:
Individual predictors can be tested using the t-statistic.
BIAS IN LINEAR MODELS
An outlier is a case that differs substantially from the main trend in the data. Standardized residuals can be used to check which residuals are unusually large and can be viewed as an outlier. Standardized residuals are residuals converted to z-scores. Standardized residuals greater than 3.29 are considered an outlier (1), if more than 1% of the sample cases have a standardized residual of greater than 2.58, the level of error in the model may be unacceptable (2) and if more than 5% of the cases have standardized residuals with an absolute value greater than 1.96, the model may be a poor representation of the data (3).
The studentized residual is the unstandardized residual divided
.....read more Discovering statistics using IBM SPSS statistics by Andy Field, fifth edition – Summary chapter 10Researchers should not compare artificially created groups in an experiment (e.g. based on the median). There are several problems with median-splits:
CATEGORICAL PREDICTORS IN THE LINEAR MODEL
Comparing the difference between the means of two groups is predicting an outcome based on membership of two groups. A t-statistic is used to ascertain whether a model parameter is equal to zero. In other words, the t-statistic tests whether the difference between group means is equal to zero.
THE T-TEST
There are two types of t-tests:
The t-test is used to see whether there is an actual difference between two groups (e.g. experimental and control). If there is no difference between the two groups, we expect to see the same mean. There is natural variation in each sample, so the mean is (almost) never exactly the same. Therefore, just by looking at the means, it is impossible to state whether there is a significant difference between two groups. In the t-test, a set level of confidence (normally 0.95), alpha, is used as a threshold of when the difference is significant. The t-statistic is used to compute a p-value and this p-value is compared to the alpha. If the p-value is equal to or smaller than the alpha, it means that there is a significant difference between the two means and then we state that there is an actual difference. The larger the difference between two means relative to the standard error, the more likely it is that there is an actual difference between the two means.
The t-test is always computed under the assumption that the null hypothesis is true. It uses the following general formula:
The null hypothesis usually states that there is no difference between the two means, meaning that the null hypothesis mean would equal ‘0’. The standard error of the sampling distribution is the standard error of differences. The standard error helps the t-test because it gives a scale of likely variability between samples.
The variance sum law states that the variance of a difference between two independent variables is equal to the sum of their variances (e.g. the variance of x1-x2 = variance of x1 + variance x2). The variance of the sampling distribution of difference between two sample means is equal to the sum of variances of the two populations from which the samples were taken. This leads to the following formula for the standard error:
This equation holds if the
.....read more Discovering statistics using IBM SPSS statistics by Andy Field, fifth edition – Summary chapter 11Moderation refers to the combined effect of two or more predictor variables on an outcome. This is also known as an interaction effect. A moderator variable is one that affects the relationship between two others. It affects the strength or direction of the relationship between the variables.
The interaction effect indicates whether moderation has occurred. The predictor and the moderator must be included for the interaction term to be valid. If, in the linear model, the interaction effect is included, then the individual predictors represent the regression of the outcome on that predictor when the other predictor is zero.
The predictors are often transformed using grand mean centring. Centring refers to transforming a variable into deviations around a fixed point. This fixed point is typically the grand mean. Centring is important when the model contains an interaction effect, as it makes the bs for lower-order effects interpretable. It makes interpreting the main effects easier (lower-order effects) if the interaction effect is not significant.
The bs of individual predictors can be interpreted as the effect of that predictor at the mean value of the sample (1) and the average effect of the predictor across the range of scores for the other predictors (2) when the variables are centred.
In order to interpret a (significant) moderation effect, a simple slopes analysis needs to be conducted. It is comparing the relationship between the predictor and outcome at low and high levels of the moderator. SPSS gives a zone of significance. Between two values of the moderator the predictor does not significantly predict the outcome and below and above the values it does.
The steps for moderation are the following if there is a significant interaction effect: centre the predictor and moderator (1), create the interaction term (2), run a forced entry regression with the centred variables and the interaction of the two centred variables (3).
The simple slopes analysis gives three models. One model for a predictor when the moderator value is low (1), one model for a predictor when the moderator value is at the mean (2) and one model for a predictor when the moderator value is high (1).
If the interaction effect is significant, then the moderation effect is also significant.
MEDIATION
Mediation refers to a situation when the relationship between the predictor variable and an outcome variable can be explained by their relationship to a third variable, the mediator. Mediation can be tested through three linear models:
There are four conditions for mediation: the predictor variable must significantly predict the outcome variable (in model 1)(1), the predictor variable must significantly predict the mediator
.....read more Discovering statistics using IBM SPSS statistics by Andy Field, fifth edition – Summary chapter 12The overall fit of a linear model is tested using the F-statistic. The F-statistic is used to test whether groups are significantly different and then specific model parameters (the bs) are used to show which groups are different.
The F-statistic gives an associated p-value as well. A p-value which is smaller than 0.05 (or any set alpha) stands for a significant difference between the group means. The downside of the F-test is that it does not tell us which groups are different. Associated t-tests can show which groups are significantly different.
The null hypothesis if the F-statistic is that the group means are equal and the alternative hypothesis is that the group means are not equal. If the null hypothesis is true, then the b-coefficients should be zero. The F-statistic can also be described as the ratio of explained to the unexplained variation.
The total sum of squares is the total amount of variation within the data. This can be calculated by using the following formula:
It is the difference between each observed data point and the grand mean squared. The grand variance is the total sum of squares of all observations. It is the variation between all scores, regardless of the group from which the scores come.
The model sum of squares is calculated by taking the difference between the values predicted by the model and the grand mean. It tells us how much of the variation can be explained using the model. It uses the following formula:
It is the difference of the group mean and the grand mean squared. This value is multiplied with the number of participants in this group and these values for each group are added together.
The residual sum of squares tells us how much of the variation cannot be explained by the model. It is calculated by looking at the difference between the score obtained by a person and the mean of the group to which the person belongs. It uses the following formula:
It is the squared difference between the participant’s score (xig) and the group mean and this is done for all the participants in all the groups. The residual sum of squares can also be denoted in the following way:
One other way of denoting the residual sum of squares is the following formula:
It is the variance of a group multiplied by one less than the number of people in that group and this value is added together for all the groups. The average sum of squares (mean squares) is calculated by dividing the model sum of squares with the degrees of freedom (N-k).
ASSUMPTIONS WHEN COMPARING MEANS
There are several assumptions when
Discovering statistics using IBM SPSS statistics by Andy Field, fifth edition – Summary chapter 13Covariates are characteristics of the participants in an experiment. These are characteristics outside of the actual treatment. If a researcher wants to compare means of multiple groups using the additional predictors, the covariates, then the ANCOVA is used. Examples of covariates could be love for puppies, softness of puppy fur.
Covariates can be included in an ANOVA for two reasons:
If there are covariates, the b-values represent the differences between the means of each group and the control adjusted for the covariate.
ASSUMPTIONS AND ISSUES WITH ANCOVA
There are two new assumptions for ANCOVA that are not present with ANOVA. These assumptions are independence of the covariate and treatment effect and homogeneity of regression slopes.
The ideal case is that the covariate is independent from the treatment effect. If the covariate is not independent from the treatment effect, then the covariate will reduce the experimental effect because it explains some of the variance that would otherwise be applicable to the experiment. The ANCOVA does not control for or balance out the differences caused by the covariate. The problem of covariates potentially explaining a bit of the data and wanting to filter these confounds is using randomizing participants to experimental groups or matching experimental groups on a covariate.
Another assumption of the ANCOVA is that the relationship between covariate and outcome variable holds true for all groups of participants and not only for a few groups of participants (e.g. for both males and females and not only males). This assumption can be checked by checking the regression line for all the covariates and all the conditions. The lines should be similar.
In order to test the assumption of homogeneity of regression slopes, the ANCOVA model should be customized on SPSS to look at the independent variable x the covariate interaction.
CALCULATING THE EFFECT SIZE
The partial eta squared is the effect size which takes the covariates into account. It uses the proportion of variance that a variable explains that is not explained by other variables in the analysis. It uses the following formula:
Discovering statistics using IBM SPSS statistics by Andy Field, fifth edition – Summary chapter 14Factorial designs are used when there are more than one independent variables. There are several factorial designs:
INDEPENDENT FACTORIAL DESIGNS AND THE LINEAR MODEL
The calculation of factorial designs is similar to that of ANOVA, but the explained variance (between-groups variance) consists of more than one independent variable. The model sum of squares (between-groups variance) consists of the variance due to the first variable, the variance due to the second variable and the variance due to the interaction between the first and the second variable.
It uses the following formula:
This is the model sum of squares and shows you how much variance the independent variables explain. It can be useful to see how much of the total variance each independent variable explains. This can be done by using the same formula, but then only for one independent variable. In order to achieve this, the independent variable has to be grouped together in one group (this normally increases the n, as more multiple groups are being put together in one big group).
The residual sum of squares, the error variance (SSR) shows how much variance cannot be explained by the independent variables. It uses the following formula:
It is the variance of a group times the number of participants in the group minus one for each group added together. The degrees of freedom are added up together too. In a two-way design, the F-statistic is computed for the two main effects and the interaction.
OUTPUT FROM FACTORIAL DESIGNS
A main effect should not be interpreted in the presence of a significant interaction involving that main effect. In other words, main effects don’t need to be interpreted if an interaction effect involving that variable is significant.
Simple effects analysis looks at the effect of one independent variable at individual levels of the other independent variable. When judging interaction graphs, there are two general rules:
Discovering statistics using IBM SPSS statistics by Andy Field, fifth edition – Summary chapter 15Repeated-measures refers to when the same entities (e.g. people) participate in all conditions of an experiment or provide data at multiple points of time.
One of the assumptions of the standard linear model is that residuals are independent, which is not true for repeated-measures designs. The residuals are affected by both between-participant factors and within-participant factors. There are two solutions to this:
One of these assumptions is sphericity (circularity). This assumption states that the relationship between scores in pairs of treatment conditions is similar (e.g. the level of dependence between means is roughly equal). It states that variation within conditions are similar and that no two conditions are any more dependent than any other two. Local sphericity refers to when some conditions do have equal variance and some do not. Sphericity is not relevant if there are only two groups. It becomes relevant when there are at least three conditions.
The assumption of sphericity can be tested using Mauchly’s test. The degree of sphericity can be estimated using the Greenhouse-Geisser estimate or the Huyn-Feldt estimate. If the assumption of sphericity is not met, then there is a loss of power and the F-statistic doesn’t have the distribution it is supposed to have. In order to do post-hoc tests when you worry about whether the assumption of sphericity is violated, Bonferonni method can be used, if it is not violated, Tukey’s test can be used.
If the assumption of sphericity is violated, the degrees of freedom has to be adjusted. The degrees of freedom is multiplied by the estimate of sphericity to calculate the adjusted degrees of freedom.
F-STATISTIC OF REPEATED MEASURES DESIGN
In repeated measured designs, the within-groups variance consists of within-participant variance, as there is only one group of participants. This consists of the effect of the experiment and the error (variance not explained by the experiment). The between-groups variance now consists of the between-participant variance.
The formula for the within-entity (groups) variance is the following:
The n represents the number of scores within the person (e.g. number of experimental conditions). The total amount of variance that is explained by the experimental manipulation can be calculated by comparing the condition mean to the grand mean for all the conditions. It uses the following formula:
The total error variance (residual sum of squares), the amount of variance that cannot be explained by the experimental manipulation can be calculated in the following way:
In order to calculate the F-statistic, the mean squares have to be calculated and this can be done by dividing both the SSR and the SSM by the degrees of freedom:
Discovering statistics using IBM SPSS statistics by Andy Field, fifth edition – Summary chapter 16Mixed designs are a combination of repeated-measures and independent designs. It includes some independent variables that were measured using different entities and some independent variables that used repeated measures.
The most important assumptions of the mixed designs ANOVA are sphericity and homogeneity of variance.
Discovering statistics using IBM SPSS statistics by Andy Field, fifth edition – Summary chapter 17A multivariate analysis is used when there is more than one dependent (outcome) variable. It is possible to use several F-tests when there are several dependent variables, but this inflates the type-I error rate. A MANOVA can detect whether groups differ along a combination of dimensions. MANOVA has a greater potential power to detect an effect.
A matrix is a grid of numbers arranged in columns and rows. The values within a matric are called components or elements and the rows and columns are vectors. A square matrix has an equal number of columns and rows. An identity matrix is a matrix where the diagonal numbers are ‘1’ and the non-diagonal numbers are ‘0’. The sum of squares and cross-products (SSCP) matrices are a way of operationalize multivariate versions of the sums of squares. The matrix that represents the systematic variance (model sum of squares) is denoted by the letter ‘H’ and is called the hypothesis sum of squares and cross-products matrix (hypothesis SCCP). The matrix that represents the unsystematic variance (residual sum of squares) is denoted by the letter ‘E’ and is called the error sums of squares and cross-products matrix (error SSCP). The matrix that represents the total sums of squares for each outcome (total SSCP) is denoted by the letter ‘T’.
The cross-product is the total combined error between two variables.
THEORY BEHIND MANOVA
The total sum of squares is calculated by calculating the difference between each of the scores and the mean of those scores, then squaring those differences and adding them together.
The degrees of freedom is N-1. The model sum of squares is calculated by taking the difference between each group mean and the grand mean, squaring it, multiplying by the number of scores in the group and then adding it all together.
The degrees of freedom is the sample size of each group minus one multiplied by the number of groups. The SST and the SSM then have to be divided by their own degrees of freedom, before being divided by each other to get to the F-statistic.
The cross-product is the difference between the scores and the mean for one variable multiplied by the difference between the scores and the mean for another variable. It is similar to covariance. It uses the following formula:
For each outcome (dependent) variable, the score is taken and subtracted from the grand mean for that variable. This gives x values per participant, with x being the number of outcome variables.
The model cross-product, how the relationship between the outcome variables is influenced by the experimental manipulation, uses the following formula:
The residual cross-product, how the relationship between the outcome variables is influenced by individual differences and unmeasured variables, can be calculated
.....read more Discovering statistics using IBM SPSS statistics by Andy Field, fifth edition – Summary chapter 18Factor analysis and principal component analysis (PCA) are techniques for identifying clusters of variables. These techniques have three uses: understanding the structure of a set of variables (1), construct a questionnaire to measure an underlying variable (2) and reduce a dataset to a more manageable size while retaining as much of the original information as possible (3).
Factor analysis attempts to achieve parsimony by explaining the maximum amount of common variance in a correlation matrix using the smallest number of explanatory constructs (latent variables). PCA attempts to explain the maximum amount of total variance in a correlation matrix by transforming the original variables into linear components.
A factor loading refers to the coordinate of a variable along a classification axis (e.g. Pearson correlation between factor and variable). It tells us something about the relative contribution that a variable makes to a factor.
In factor analysis, scores on the measured variables are predicted from the means of those variables plus a person’s scores on the common factors (e.g. factors that explain the correlations between variables) multiplied by their factor loadings, plus scores on any unique factors within the data (e.g. factors that cannot explain the correlations between variables).
In PCA, the components are predicted from the measured variables.
One major assumption of factor analysis is that the algebraic factors represent real-world dimensions. A regression technique can be used to predict a person’s score on a factor. Using this technique, the resulting actor scores have a mean of 0 and a variance equal to the squared multiple correlation between the estimated factor scores and the true factor values. A downside is that the scores can correlate with other factor scores from a different orthogonal factor. The Bartlett method and the Anderson-Rubin method can be used to overcome this problem. The Bartlett method produces factor scores that are uncorrelated and standardized.
DISCOVERING FACTORS
The method used for discovering factors depends on whether the results should be generalized from the sample to the population (1) and whether you are exploring your data or testing a specific hypothesis (2).
Random variance refers to variance that is specific to one measure but not reliably so. Communality refers to the proportion of common variance present in a variable. Extraction refers to the process of deciding how many factors to keep.
Eigenvalues associated with a variate indicate the substantive importance of that factor. Therefore, factors with large eigenvalues are retained. Eigenvalues represent the amount of variation explained by a factor.
A scree plot is a plot where each eigenvalue is plotted against the factor with which it is associated. The point of inflexion is where the slope of the line changes dramatically. This point can be used as a cut-off point to retain factors. It is also possible to use eigenvalues as a criterion. Kaiser’s criterion is to retain factors with eigenvalues greater than 1. Joliffe’s criterion
.....read more Discovering statistics using IBM SPSS statistics by Andy Field, fifth edition – Summary chapter 19It is possible to predict categorical outcome variables, meaning, in which category an entity falls. When looking at categorical variables, frequencies are used. The chi-squared test can be used to see whether there is a relationship between two categorical variables. It is comparing the observed frequencies with the expected frequencies. The chi-squared test standardizes the deviation for each observation and these are added together.
The chi-squared test uses the following formula:
The expected score has the following formula:
The degrees of freedom of the chi-squared distribution are (r-1)(c-1). In order to use the chi-squared distribution with the chi-squared statistic, there is a need for the expected value in each cell to be greater than 5. If this is not the case, then Fisher’s exact test can be used.
The likelihood ratio statistic is an alternative to the chi-square statistic. It is comparing the probability of obtaining the observed data with the probability of obtaining the same data under the null hypothesis. The likelihood ratio statistic uses the following formula:
It uses the chi-squared distribution and is the preferred test if the sample size is small. The chi-square statistic tends to make a type-I error if the table is 2 x 2. This can be corrected for by using Yates’ correction and uses the following formula:
In short, the chi-square test tests whether there is a significant association between two categorical variables.
ASSUMPTIONS WHEN ANALYSING CATEGORICAL DATA
One assumption the chi-square test uses is the assumption of independence of cases. Each person, item or entity must contribute to only one cell of the contingency table. Another assumption is that in 2x2 tables, no expected value should be below 5. In larger tables, not more than 20% of the expected values should be below 5 and all expected values should be greater than 1. Not meeting this assumption leads to a reduction in test power.
The residual is the error between what the expected frequency and the observed frequency. The standardized residual can be calculated in the following way:
Individual standardized residuals have a direct relationship with the test statistic, as the chi-square statistic is composed of the sum of the standardized residuals. The standardized residuals behave like z-scores.
EFFECT SIZE
Cramer’s V can give an effect size. In 2x2 tables, the odds-ratio is often used as the effect size. The odds-ratio uses the following formula:
The actual odds ratio is the odds of event A divided by the odds of event B.
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