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: 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 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:

1. How big the effect is
2. The alpha
3. The sample size

There are some rules of thumb about confidence intervals:

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

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