# Inferential Statistics, Howell Chapter 4,5,6

Inferential Statistics, Howell Chapter 4,5,6

**Sampling error: **

Also “chance variability”. Variability in findings are due to chance

**Hypothesis testing: **

Reason: Data are ambiguous à means are different

Goal: Find out if the difference is big or small i.o.w à statistically significant

**Sampling distributions: **

What degree of variability of sample-sample can we expect in the data?

Tells us what variability we can expect under certain conditions (e.g. if population mean is equal).

Can also be done with other measure of variability: Range,

**Sampling distribution of differences between means:**

Compares distribution of __means__

**Standard error: **

Expected standard deviation of samples of measured statistic, when measured repeatedly.

Theory of Hypothesis Testing

- Answering statistical significance is no longer sufficient (p<.05)

à Need to inform reader about **power **and **confidence limits **and **effect size**

** - **Try to find out if difference in sample means **(sampling distribution) **is likely if the sample was drawn from a population with an equal mean

Process:

1. Set up the **research hypothesis. **Eg. Parking takes longer if someone watches

2. Collect random sample under the 2 conditions

3. Set up Ho = **null hypothesis **= the population means of the 2 samples are equal

4. Calculate **sampling distribution **of the 2 means under condition that Ho is true

5. Calculate probability of a mean difference that is *at least as large* as the one obtained

6. Reject or fail to reject Ho (Assumption that Ho is not true – not proven !!!!)

1. Research Hypothesis

2. Collect random sample

3. Set up null hypothesis

4. Sampling distribution under Ho=true

5. Compare sample statistic to distribution

6. Reject or retain Ho

**Null hypothesis:
**- Usually the opposite of the research hypothesis

à in order to be disproven (cause we can never prove something, only disprove the

opposite.

Statistical conclusions

- Fisher:

- Options are to **reject **or **suspend **judgement over Ho.

à If Ho cannot be rejected, the judgement about it has to be suspended.

(eg. Schoolexperiment continues)

- Neyman-Pearson:

- Options are to **reject **or **accept **that Ho is true.

à If Ho cannot be rejected, Ho has to be considered true until disproven.

(eg. Schoolexperiment stops, until evidence has to be reconsidered)

**Conditional Probabilites:
**- Confusion between the probability of the hypothesis given the data and the data given the hypothesis.

à p = .045 means that probability of data given if hypothesis Ho = true à p(D I Ho)

Test Statistics

**Sample statistics: **

- Descriptives (mean, range, variance, correlation coefficient)

- Describe characteristics of the samples

**Test statistics: **

- Statistical procedure with own sampling distributions (t, F , X²)

Decisions about the Null-Hypothesis

**Rejection level / significance level:**

- Sample score falls inside the 5% level of the assumed distribution à **rejection region**

à If it falls there, the likelihood that the findings are due to chance is 5%

à Therefore it is **statistically significant**

Type I and Type II Errors

**Type I : (Jackpot Error)**

- Rejecting Ho when it is actually true

à Probability of making this error is expressed as **alpha
**

**à**We will make this error 5% of the time

** Type II :
- **Fail to reject Ho when it is actually wrong

à Probability of making this error is expressed as

**beta**

**à**We will make this error depending on the size of

**rejection region**

- Less Type I error = more Type II error

**Power: **

If **beta **is smaller, the distance between sample mean and pop mean is bigger, thus the generalizability increases. à More power

One and Two Tailed Tests

**One tailed / directional test:
- **test only for one direction of the distribution 5% level

**Two tailed / nondirectional test:
- **test for negative and positive scores on 2.5% level

- Reasons: No clue what data will look like

Cover themselves in the event the prediction was wrong

One tailed tests are hard to define (if more than two groups)

àTry to keep statistical significance low.

2 Questions to deal with any new statistic

1. How and with what assumption is the statistic calculated?

2. What does the statistic´s sampling distribution look like under Ho?

à compare

Alternative view of hypothesis testing

Traditional way:

- Null hypothesis = m1 = m2 or m1 not = m2 (two tailed)

According to Jones, Tukey and Harris

- 3 possible conclusions

1. m1 < m2

2. m1 > m2

3. m1 = m2

- 3. Is ruled out, because the means are never the same. So we test for 2 directions at the same time. It allows us to keep 5% levels at both ends of the distribution, because we will just discard the other one

Basic Concepts of Probability

1.2 Basic Terminology and Rules. 1

2.0 Discrete vs Continuous Variables. 2

**Analytic view: **Common definition of probability. Even can occur in A ways and fail to occur in B ways.

à all possible ways are equally likely (definite probability, eg. 50%)

** **Probability of occurrence: A/(A+B) à p(blue)

Probability of failure to occur: B/(A+B) à p(green)

**Frequentist view: **Probability is the limit of the relative frequency of occurrence

à Dice will land approx. 1/6^{th} of time on one side with multiple throws (proportions)

**Subjective probability: **Individuals subjective estimate. (opposite of frequentist view)

à use of Bayes´ theorem

à usually disagree with general hypothesis testing orientation

1.2 Basic Terminology and Rules

**Event: **The occurrence of “something”

**Independent event: **Set of events that do not have an effect on each others occurrences

**Mutually exclusive event: **The occurrence of one event precluded the occurrence of the alternative event.

**Exhaustive event: **All possible occurences /outcomes (e.g. die) are considered.

**Theorem: **Rule

**(Sampling with replacement: **Before drawing a new sweet (occurrence), the old draw is replaced.)

**Additive law of probability: **(__mutually exclusive__ event must be given)

The occurrence of one event is equal to the sum of their separate probabilities.

p(blue or green) = p(blue) + p(green) = .24 + .16 = .40

à one outcome (occurrence)

**Mulitplicative Rule: **(__independence__ of events must be given)

Probability of their joint (successive/co-occurrence) occurrence is product of individual

probabilities.

p(blue, blue) = p(blue) * p (blue) = .24 * .24 = .0576

à minimum 2 outcomes (occurrences)

**Joint probability: **Probability of the co-occurrence of two or more events

- If **independent, **p can be calculated with **multiplicative law **

- If not independent, than very complicated procedure (not given in book)

Denoted as: p(A, B) à p(blue, green)

**Conditional probability: **Probability an event occurs **if / given **another event has occurred.

à hypothesis testing: **If **Ho = true, the **p** of this result is….

à Conditional can be read as: If…is true, then

Denoted as: p (A I B) à p(Aids I drug user)

2.0 Discrete vs Continuous Variables

D**iscrete variable: **Can take on specific values à 1,2,3,4,5

à Probability distribution:

Proportions translate directly to probability

à can be measured at ordinate (Y-axis) – relative frequency

**Continuous variable: **Can take on infinite values à 1.234422 , 2.234 , 4 …

à Variable in experiment can be considered

continuous if min. **ordinal scale **(e.g. IQ)

**Density: **height of the curve at point X

à Probability distribution:

Likelihood of one specific score is not useful, cause p(X = exactly,

e.g. 2) is highly unlikely, rather 2.1233

à Measure **Interval**: E.g. 1.5 – 2.5

à Area under defined interval, a to b = our probability à use distribution tables (later chapters)

Inhalt

6.0 Basics for Chi-Square tests. 1

6.1 Chi-Square Distribution. 1

6.2 Chi-Square Goodness of Fit Test – One-way Classification. 2

6.2.1 Tabled Chi-Square Distribution. 3

6.3 Two Classification Variables: Contingency Table Analysis. 3

6.3.2 Correcting for Continuity (for 2 x 2 tables + expected frequency is small). 4

6.3.3 Fischers Exact Test (another test, besides the chi-square test). 4

6.12 Kappa - Measure of Agreement. 4

6.13 How to write down findings – see book !!!!. 5

# 6.0 Basics for Chi-Square tests

**Measurement data: **(also quantitative data): Observation represents score on a continuum (e.g. mean, st. dev.)

**Categorical data: **(also frenquency data): Data consists of frequencies of observations that fall into 2 or more

categories. à remember frequency tables

**Chi-square X²:** 2 different meanings: 1. Mathematical distribution that stands for itself

**or Pearson´s chi-square** 2. Refers to a statistical test of which the result is distributed in

approximately the same way as ** X²**

** Assumptions of Chi-square test: **Observations need to be **independent** of each other

+ Aim is to **test independence of variables** (significance of findings)

6.1 Chi-Square Distribution

**Chi-square Distribution:
**

** Explanation:
Gamma function: = **factorial.

When argument of gamma (k/2), then gamma = integer à [(k/2) – 1]!

à Need of gamma functions because arguments not always integers

- Chi-square has only one parameter **k. **(≠ two-parameter functions with µ and ơ )

- Everything else is either a **constant e **or another value of **X²**

(- **X²3 ** is read as “chi-square with 3 degrees of freedom = **df **(expl. Later))

# 6.2 Chi-Square Goodness of Fit Test – One-way Classification

**Chi-square test: **- based on **X² **distribution.

- can be used for one-dimensional tables and two-dimensional (contingency tables)

**!!!! Beware: **We need large expected frequencies: **X² distribution **is** continuous** and cannot provide a good

approximate if we have only a few possible Efrequencies, which are **discrete.**

**à Should minimum be: Efreq. ≥ 5 ,**otherwise **low power to reject Ho**.

(e.g. flipping a coin only 3 times cannot be compared with the frenquency distribution because the

frequency is just too small) – It could be compared but this is stupid :P

**nonoccurences: **Have to be mentioned in the table. Cannot compare 2 variables that only show

one observation.

**Goodness-of-fit test: **Test whether difference of observed score from expected scores are big enough to

question whether this is by chance or significant. **Significance test** or **Independence test**.

**observed frequency: **Actual data collected

**expected frequency: **Frequency expected if **Ho **were true.

6.2.1 Tabled Chi-Square Distribution

We have obtained a value for X² and now we have to compare it to the **X² distribution **to get a probability,

so we can define whether our X² is significant (reject Ho) or we accept our H1.

For this we use: **Tabled distribution of X²:
**depends on

**df = degrees of freedom**

**à df = k-1**(number of categories -1)

# 6.3 Two Classification Variables: Contingency Table Analysis

** **We want to know if a variable is **contingent **or **conditional on ** a second variable.

We do this by using a

**contingency table: **

**Marginal total: (**Rowtotal * Columntotal) – N

See also: Formula for joint occurrence of independent events (chapter 5)

Now continue with calculation of the **chi-square** to determine **significance** of findings.

Now, to assess whether our X² is significant, we first have to calculate the **degree of freedom =df ** to know

where to look on the **X² distribution table**

** **

6.3.2 Correcting for Continuity (for 2 x 2 tables + expected frequency is small)

**Yate´s correction for continuity: **Reducing absolute value of each numerator (O-E) for 0.5 before squaring

6.3.3 Fischers Exact Test (another test, besides the chi-square test)

**Fischer´s Exact Test: **Is mentioned, but I think not exam material. If it is, I will update the summary.

# 6.12 Kappa - Measure of Agreement

**Kappa ( k ) : **Statistic that measures interjudge agreement by using contingency tables (not based on chi-square) à measure of reliability

à corrects for chance

1. First calculate expected frequencies for diagonal cells = (cells in which the judges agree = relevant)

2. Apply formula. Result = k Kappa

# 6.13 How to write down findings – see book !!!!

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