## Summary of Discovering statistics using IBM SPSS statistics by Andy Field - 5th edition

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Discovering statistics using IBM SPSS statistics by Andy Field, fifth edition – Summary chapter 2

Many statistical models try to predict an outcome from one or more predictor variables. Statistics includes five things: **S**tandard error (**S**), **p**arameters (**P**), **i**nterval estimates (**I**), **n**ull hypothesis significance testing (**N**) and **e**stimation (**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 **are boundaries within which we believe the population value (*parameter*) will fall. In an **interval estimate**, the sample value is used as a midpoint and a lower- and upper limit is set as well.

If the sample is large enough it is possible to assume normal distribution. The values used for a 95% confidence interval are then -1,96 and 1,96, because a probability of 0,0250 is associated with a z-score of 1,96. This gives the following formula:

The smaller the interval, the closer the true mean is to the sample mean. It is not possible to make probability statements about confidence intervals. If a normal distribution cannot be assumed, then the t-distribution is used. A confidence interval using the t-distribution uses the following formula:

‘n-1’ refers to the degrees of freedom of the sample, which is sample size, minus one. The degrees of freedom and the desired confidence level together determines the t that is used.

__NULL HYPOTHESIS SIGNIFICANCE TESTING__

A p-value of 0,05 or smaller is generally used in science. In order to assess a claim, the probability of an event has to be calculated and if the probability is 0,05 or smaller, then we do not reject the claim.

The **alternative hypothesis **is the hypothesis that there is an effect or a difference. The **null hypothesis **states that there is no effect or difference. It is not possible to accept the null hypothesis. If the alternative hypothesis is rejected, then the null hypothesis is not accepted, but seems plausible. If the null hypothesis is rejected in favour of the null hypothesis, then the alternative hypothesis is not accepted, but seems plausible. Whether the alternative hypothesis is rejected or not depends on the p-value that is set and found in the sample.

A **type-I error **is falsely rejecting the null hypothesis. The probability of the type-I error is equal to the set alpha. The **alpha** is the set value of when to reject the hypothesis (*e.g. a=0.05*). Significance testing makes use of the t-distribution. The **power** of the test is how much data needs to be collected to have a reasonable chance of finding the effect you’re looking for. In other words, it is the needed sample size. It is the ability of a test to find an effect, given that one exists. The best tests have great power and a small probability of a type-I error.

With significance testing, the p-value is calculated using the **test-statistic**. This is the ratio of effect relative to error and uses the basic formula:

There are a lot of different formulas for it, but it is what is being used to calculate the p-value. If the p-value is below the set alpha, the test statistic is said to be **statistically significant**, as it is presumed that the test statistic reflects a genuine effect in the real world.

A significance test can be one-sided or two-sided. A non-directional hypothesis (*two-sided*) is a **two-tailed test** and a directional hypothesis (*one-sided*) is a one-tailed test. If a one-tailed test is in the opposite direction of what you have expected, you cannot reject the null hypothesis.

A **type-II error **refers to believing there is no effect in the population when there is, in other words, it is falsely rejecting the alternative hypothesis. The probability of a type-II error is also called the b-level. The **familywise **or **experimentwise error rare **is the greater probability of a type-I error if several tests are used. It uses the following formula:

This increased probability of a type-I error can be fixed by increasing the significance level of all tests so that the overall probability of a type-I error remains at 0,05. This can be done by using the following formula:

It is dividing the significance level by the number of comparisons (k). The power of a test depends on several things:

**How big the effect is****The alpha****The sample size**

There are some rules of thumb about confidence intervals:

- If the ends touch, it represents a p-value for testing the null hypothesis of no differences of 0.01.
- If there is a gap between the upper end of one and the lower end of another then p<0.01.
- A p-value of 0.05 is represented by a moderate overlap.

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Discovering statistics using IBM SPSS statistics by Andy Field, fifth edition – Summary chapter 1

The **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:

**Binary variable**

A categorical variable with two options (*e.g. ‘yes’ or ‘no’*).**Nominal variable**

A categorical variable with more than two options (*e.g. hair colour*).**Ordinal variables**

A categorical variable that has been ordered (*e.g. winner and runner-up*)

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

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Discovering statistics using IBM SPSS statistics by Andy Field, fifth edition – Summary chapter 2

Many statistical models try to predict an outcome from one or more predictor variables. Statistics includes five things: **S**tandard error (**S**), **p**arameters (**P**), **i**nterval estimates (**I**), **n**ull hypothesis significance testing (**N**) and **e**stimation (**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**

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Discovering statistics using IBM SPSS statistics by Andy Field, fifth edition – Summary chapter 3

There are three main misconceptions of statistical significance:

**A significant result means that the effect is important**

Statistical significance is not the same as practical significance.**A non-significant result means that the null hypothesis is true**

Rejecting the alternative hypothesis does not mean we accept the null hypothesis.**A significant result means that the null hypothesis is false**

If we reject the null hypothesis in favour of the alternative hypothesis, this does not mean that the null hypothesis is false, as rejection is all based on probability and there still is a probability of it not being false.

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: **E**ffect sizes (**E**), **M**eta-analysis (**M**), **B**ayesian **E**stimation (**BE**), **R**egistration (**R**) and **S**ense (**S**), together making **EMBERS**.

__SENSE__

There are six principles for when using NHST in order to use your sense:

- The exact p-value can indicate how incompatible the data are with the null hypothesis.
- P-values are not interpreted as the probability that the hypothesis is true.

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Discovering statistics using IBM SPSS statistics by Andy Field, fifth edition – Summary chapter 5

A good graph has the following properties:

- Shows the data
- Induce the reader to think about the presented data
- Avoid distorting data
- Present many numbers with minimum ink
- Make large data sets coherent
- Encourage the reader to compare different pieces of data
- Reveal the underlying message of data

There are also some graph building guidelines:

- If plotting two variables, never use 3-D plots
- Do not use unnecessary patterns in the bars
- Do not use cylinder shaped bars if that is not functional
- Properly label the x- and y-axis.

__HISTOGRAMS__

There are different types of histograms:

**Simple histogram**

Visualize frequencies of scores for a single variable**Stacked histogram**

Compare relative frequencies of scores across groups**Frequency polygon**

The same as a simple histogram, but uses a line, instead of a bar**Population pyramid**

Comparing distributions across groups and the relative frequencies of scores in two populations.

__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:

**1-D boxplot**

A single boxplot for all scores of the chosen outcome**Simple boxplot**

Multiple boxplots for the chosen outcome by splitting the data by a categorical variable**Clustered boxplot**

A simple boxplot, but it splits the data by a second categorical variable.

__BAR CHARTS__

**Bar charts **are often used to display means. There are different types of bars:

**Simple bar**

The means of scores across different groups or categories.**Clustered bar**

Different coloured bars to represent levels of a second grouping variable (*e.g: film rating and excitement and enjoyment*)**Stacked bar**

Clustered bar, but the bars are stacked.**Simple 3-D bar**

Second grouping variable is represented by an additional axis**Clustered 3-D bar**

A clustered bar, but an extra categorical variable can be added on an extra axis**Stacked 3-D bar**

A 3-D clustered bar, but the bars are stacked**Simple error bar**

A simple bar chart, but there is no bar, but a line and a dot**Clustered error bar**

Clustered bar chart, but a dot with an error band around it.

__LINE CHARTS__

**Line charts **are bar charts but with lines instead of bars. There are two types of line charts:

**Simple line**

The means of scores across different groups of cases**Multiple line**

This is equivalent to the clustered bar chart.

__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:

**Simple scatter**

A scatterplot of

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Discovering statistics using IBM SPSS statistics by Andy Field, fifth edition – Summary chapter 6

Bias 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:

- Additivity and linearity

The scores on the outcome variable are linearly related to any predictors. If there are multiple predictors, their combined effect is best described by adding them together. - Normality

The parameter estimates are influenced by a violation of normality and the residuals of the parameters should be normally distributed. It is normality for each level of the predictor variable that is relevant. Normality is also important for confidence intervals and for null hypothesis significance testing. - Homoscedasticity / homogeneity of variance Homoscedasticity / homogeneity of variance

This impacts the parameters and the null hypothesis significance testing. It means that the variance of the outcome variable should not change between levels of the predictor variable. Violation of this assumption leads to bias in the standard error. - Independence

This assumption means that the errors in the model are not related to each other. The data has to be independent.

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:

**Trim the data**

Delete a

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Discovering statistics using IBM SPSS statistics by Andy Field, fifth edition – Summary chapter 7

**Non-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:

**Kolmogorov-Smirnov Z**

It tests whether two groups have been drawn from the same population. It has more power than the Mann-Whitney test when the sample sizes are less than 25 per group.**Moses Extreme Reaction**

It tests the variability of scores across the two groups and is a non-parametric form of the Levene’s test.**Wald-Wolfowitz runs**

It looks at clusters of scores in order to determine whether the groups differ. If there is no difference, the ranks should be randomly interspersed.**Sign test**

It does the same as the Wilcoxon-signed rank test but it is only based on the direction of the difference. The magnitude of change is neglected. It lacks power unless the sample size is really small.**McNemar’s test**

It uses nominal, rather than ordinal data. It is useful when looking for changes in people’s scores. It compares the number of people who changed their response in one direction to those who changed in the opposite direction.**Marginal homogeneity**

It is an extension of McNemar’s test and is similar to the Wilcoxon test.**Friedman’s 2-way ANOVA by ranks**(*k**samples*)

It is a non-parametric ANOVA to compare two groups but has low power compared to the Wilcoxon signed-rank test.**Median test**

It assesses whether samples are drawn from a population with the same median.**Jonckheere-Terpstra**

It tests for trends in the data. It tests for an ordered pattern of the medains of the group. It does the same as the Kruskal-Wallis test but incorporates the order of the groups. This test should be used when a meaningful order of medians is expected.**Kendall’s W**

It tests the agreement between raters and ranges between 0 and 1.**Cochran’s Q**

It is a Friedman test on dichotomous data.

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

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Discovering statistics using IBM SPSS statistics by Andy Field, fifth edition – Summary chapter 8

**Variance **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 **(*R**2*) is a measure of the amount of variability in one variable that is shared by the other. **Spearman’s correlation coefficient **(*r**s*) 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*

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Discovering statistics using IBM SPSS statistics by Andy Field, fifth edition – Summary chapter 9

Any 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

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Discovering statistics using IBM SPSS statistics by Andy Field, fifth edition – Summary chapter 10

Researchers should not compare artificially created groups in an experiment (*e.g. based on the median*). There are several problems with median-splits:

**Median splits change the original information drastically****Effect sizes get smaller****There is an increased chance of finding spurious effects**

__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:

**Independent t-test**(*independent measures t-test*)

This is comparing two means in which each group has its own set of participants.**Paired-samples t-test**(*dependent t-test*)

This is comparing two means in which each group uses the same participants.

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

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Discovering statistics using IBM SPSS statistics by Andy Field, fifth edition – Summary chapter 11

**Moderation **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:

- A linear model predicting the outcome from the predictor variable (
*c*). - A linear model predicting the mediator from the predictor variable (
*a*). - A linear model predicting the outcome from both the predictor variable and the mediator (
*predictor = c’*and*mediator = b*).

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

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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

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Discovering statistics using IBM SPSS statistics by Andy Field, fifth edition – Summary chapter 13

**Covariates **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:

**Reduce within-group error variance**

The unexplained variance is attributed to other variables, the covariates, which reduces the total error variance. This allows for a more sensitive test for the difference of group means.**Elimination of confounds**

By adding other variables, covariates, in the analysis, confounds are eliminated.

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:

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Discovering statistics using IBM SPSS statistics by Andy Field, fifth edition – Summary chapter 14

**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 design**

There are several independent variables using the same entities in all conditions.**Mixed design**

There are several independent variables. Some conditions use the same entities and some conditions use different entities.

__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:

- Non-parallel lines on an interaction graph indicate some degree of interaction, but how strong and whether the interaction is significant depends on how non-parallel the lines are.
- Lines on an interaction graph that cross are very non-parallel, which hints at a possible significant interaction, but does not necessarily mean that it is a significant interaction.

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Discovering statistics using IBM SPSS statistics by Andy Field, fifth edition – Summary chapter 15

**Repeated-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:

- Model within-participant variability
- Apply additional assumptions to make a simpler, less flexible model fit

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:

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Discovering statistics using IBM SPSS statistics by Andy Field, fifth edition – Summary chapter 16

**Mixed 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.

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Discovering statistics using IBM SPSS statistics by Andy Field, fifth edition – Summary chapter 17

A **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

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Discovering statistics using IBM SPSS statistics by Andy Field, fifth edition – Summary chapter 18

**Factor 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**

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Discovering statistics using IBM SPSS statistics by Andy Field, fifth edition – Summary chapter 19

It 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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