parametric statistics

Parametric statistics

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Outline

Measuring the accuracy of the mean Practical notes for practice Inferential statistics

T-testANOVA

Measuring the accuracy of the mean The mean is the simplest statistical model

that we use This statistic predicts the likely score of a

person The mean is a summry statistic

Measuring the accuracy of the mean The model we choose (mean/ median /

mode) should represent the state in the real worldDoes the model represent the world

precisely?The mean is a prefect representation only if

all the scores we collect are the same as the mean.

Mean

When the mean is a perfect fit: there is no difference between the mean and each data point

Score Child

10 1

10 2

10 3

10 4

10 5

10 6

10 7

10 8

80/8=10 Mean

Mean

Usually, there are differences between the mean and the raw scores

If the mean is representative of the data these differences are small.

Score Child

10 1

9 2

8 3

12 4

8 5

11 6

10 7

12 8

80/8=10 Mean

Deviation

The differences between the model prediction (=mean) and each raw score is the deviation

Deviation Mean Score Child

10 10 1

10 9 2

10 8 3

10 12 4

10 8 5

10 11 6

10 10 7

10 12 8

10 Mean

Deviation

Compute the deviation of each score from the mean

Measure the overall deviation (sum)

Deviation Mean Score Child

0 10 10 1

1 10 9 2

2 10 8 3

2- 10 12 4

2 10 8 5

1- 10 11 6

0 10 10 7

2- 10 12 8

0 Sum 10 Mean

Deviation

Squared dev. Deviation

0 01 14 24 2-4 21 1-0 04 2-18 0

Mean Raw score

10 10

10 9

10 8

10 12

10 8

10 11

10 10

10 12

Sum

Deviation

Squared dev. Deviation

0 01 14 24 2-4 21 1-0 04 2-18 0

Sum of squared deviations (also called the sum of squared errors) is a good measure of the accuracy of the mean

Except that it gets bigger with more scores

Variance

Divide sum of squared deviations

by the number of scores minus 1

We can compare variance across samples Square root of variance is standard deviation

Standard deviation

Variance N-1 Number of scores (N)

Sum of square deviations

1.62.57= 18/7=2.57 7 8 18

Accuracy of the mean

Sum of squared deviations (sum of squared errors), variance and standard deviation all measure the same thing: variability of the data

Standard deviation (SD) measures how well the mean represents the data: small SD indicate data points close to the mean

Standard Deviation (SD)

SD close to the meanSD Mean

0.2 7 Sentence 1

0.5 5 Sentence 2

0.1 2 Sentence 3

0.5 2 Sentence 4

0.2 5 Sentence 5

0.2 6 Sentence 6

0.1 4 Sentence 7

0.2 6 Sentence 8

Standard Deviation (SD)

SD far from the meanSD Mean

1.5 7 Sentence 1

2.5 5 Sentence 2

1.5 2 Sentence 3

2.5 2 Sentence 4

3 5 Sentence 5

1.5 6 Sentence 6

2.5 4 Sentence 7

3.5 6 Sentence 8

Why use number of scores minus 1?

We are using a sample to estimate the variance in the population

Population?

Sample-population

The intended population of psycholinguistic research can be all people / all children aged 3 / etc.

Actually, we collect data only from a sample of the population we are interested in.

We use the sample to make a guess about the linguistic behavior of the relevant population.

Sample - population

Size of the sample The mean as a model is resistant to

sampling variation: different samples from the same populations usually have a similar mean

Why use number of scores minus 1?

We are using a sample to estimate the variance in the population

Population?

Why use number of scores minus 1?

We are using a sample to estimate the variance in the population

Variance in the sample: observations can vary (5, 6, 2, 9, 3) mean=5

But: if we assume that the sample mean is the same as the population (mean=5)

Why use number of scores minus 1?

For the next sample, not all observations are free to vary.

For a sample of (5, 7, 1, 8, ?) we already need to assume that the mean is 5. (?=4)

Why use number of scores minus 1?

This does not mean we fix the value of the observation, but simply that for various statistics we have to calculate the number of observations that are free to vary.

This number is called: Degrees of freedom and it must be one less than the sample size (N-1).

To summarize

Mean represents the sample

Sample represents population

Many samples - population

Theoretically, if we take several samples from the same population

Each sample will have its own Mean and SD

If the samples are taken from the same population, they are expected to be reasonably similar.

Many samples

mean

Population 10

Sample 1 9

Sample 2 11

Sample 3 10

Sample 4 12

Sample 5 9

Sample 6 10

Sample 7 11

Sample 8 8

Frequency Mean

1 8

2 9

3 10

2 11

1 12

Sampling distribution

Frequency Mean

1 8

2 9

3 10

2 11

1 12

Average of all sample means will give the value for the population mean

Sampling distribution

How accurate is a sample likely to be?

Calculate the SD of the sampling distribution

This is called the standard error of the mean (SE)

Standard Error (SE)

We do not collect many samples, but compute SE

SE= SD/N

SD Mean

0.2 7 Sentence 1

0.5 5 Sentence 2

0.1 2 Sentence 3

0.5 2 Sentence 4

0.2 5 Sentence 5

0.2 6 Sentence 6

0.1 4 Sentence 7

0.2 6 Sentence 8

Standard Error (SE)

SE SD Mean

0.07 0.2 7

0.18 0.5 5

0.04 0.1 2

0.18 0.5 2

0.07 0.2 5

0.07 0.2 6

0.04 0.1 4

0.07 0.2 6

Why are the samples different?

Source of the variance: Different population or Sampling error (random effect, can be

calculated) Can we take results from the sample to

make generalizations about the population?

Accuracy of sample means

Calculate the boundaries within which most sample means will fall.

Looking at the means of 100 samples, the lowest mean is 2, and the highest mean is 7.

The mean of any additional sample will fall within these limits.

Confidence Interval

The limits within which a certain percent (typically we look at 95%) of sample means will fall.

If we collect 100 samples, 95 of them will have a mean within the confidence interval

PRACTICE

Experiment: Compare children with specific language

impairment (SLI) and children who are typically developing (TD).

Hypothesis: effect of word order SVO vs. VSO

Task : repeat a sentence

30 Children, each was presented with 10 sentences (5 SVO, 5 VSO).

V CORRECT?

YES/NO SVO

YES/NO VSO

VSO2 VSO1 SVO2 SVO1 gender age SLI

0 1 0 1 1 4;02 1 Child 1

0 1 1 0 1 3;04 0 Child 2

0 1 0 0 1 4;06 1 Child 3

0 1 1 1 0 3;11 0 Child 4

Compute: Mean? Frequency?

SD Mean

SVO SLI

VSO

SVO TD

VSO

Basic analysis with Excel

Descriptive statistics: Sum, Average, Percentage

Drawing graphs Parametric statistics: Mean, Standard

Deviation, t-test Smart sheets: COUNTIF

INFERENTIAL STATISTICS

Statistical hypothesis

Hypothesis for the effect of a linguistic phenomenon

Findings from a sample Do the findings support the hypothesis?

Do they show a linguistic effect? To answer this, we consider a null

hypothesis

The null hypothesis (H0)

H0= the experiment has no effect The purpose of statistical inference is to

reject this hypothesis H1 = the mean of the population affected

by the experiment is different from the general population

Rejecting the null hypothesis

Compare the mean of the sample to two populations (under H1 or H0).

We cannot show the sample belongs to the population under H1.

All we can do is compare the sample to population under H0 and consider the likelihood that it belongs to it.

Rejecting the null hypothesis

Check if our sample belongs to the population under H1 or H0

Consider confidence interval, SE Compare means Compare varience

Level of significance (alpha)

Is the difference between the sample and the population big enough to reject H0?

Determine a critical value (alpha) as criterion for including the sample in the population

< 0.05

Parametric statistics

Variables are on interval scale (at least) Compute means of raw grade (several

items to one condition)

t-testsANOVAANACOVA

t-Tests t-tests are used in order to compare two

samples and decide whether they are significantly different or not.

The t-tests represent the difference between the means of the two samples which takes into consideration the degree to which these means could differ by chance.

t-Tests The degree to which the means could

differ by chance is the Standard Error (SE) We do not calculate the t-value ourselves,

but we use it to determine the effect of the experiment on the sample.

How do we know if the t-value is significant (p<0.05)?

t-Tests

Every sample belongs to a different t-curve. This depends on the degree of freedom (df= N-1.)

Check the table of values called the Student's t-distributions, which is based on df determines. We mark the df on t.

t(32)=1.15

Types of t-tests:

Matched/Paired/Dependent t-test - compares two sets of scores for the same sample, or scores of matched samples. Sample can be of equal variance or unequal variance.

Independent (two sample) t-test - compares two different samples (on the same test). Samples can be of equal size or unequal size, of equal variance or unequal variance. The df for independent t-test is Nx-1+Ny-1.

ANOVA - Comparing Means of more than two samples ANOVA - Analysis of variance. It considers

within group variability as well as random between group variability and nonrandom between group variability.

The type of ANOVA depends on the research design - the number of independent and dependent variables.

One way ANOVA - one independent variable, one dependent variable with more than two values

Two-Way Independent ANOVA - Two independent variables, with different participants in all groups (each person contributes one score).

Two-Way Repeated Measures ANOVA - Everything comes from the same participants

Two-Way Mixed ANOVA - one independent variable is tested on the same participants, the other on different participants

F-score is the product of dividing the between group variance (which takes into consideration random and non-random variance) by the within group variance.

For every F-score we can determine the significance based on the dfs (between groups and within groups).

Post hoc comparisons

Post-hoc comparisons are used to find out where the differences which yielded significant come from.

Tukey Test - used when the sample sizes are the same.

Scheffe Test - used with unequal sample sizes, but could be used with equal sample sizes

Bonferroni correction - when there are multiple comparisons the level of significance is divided by the number of test to avoid family-wise errors.

These tests can also be used to test unplanned comparisons

ANCOVA - Analysis of covariance

Allows introducing covariates (factors other than the experimental design which might influence the results) into the ANOVA.