Computer Aided Archaeology

class: title-slide, center, middle

#  Computer Aided Archaeology

##  07 - Basic Statistics

###  Martin Hinz

####  Institut für Archäologische Wissenschaften, Universität Bern

01/11/23
---

## Flavours of statistics

### Descriptive statistics
- Summary and description of data by using parameters (mean, standard deviation etc.)

### (graphical display)
- Summary and description of data by using graphs (bar charts, pie charts etc.)
- Useful for pattern detection and description, therefore intermediate position

### Explorative statistics
- Summary and description of data for pattern detection (e.g. correspondence analysis)

### Statistical inference or statistical induction
- testing of hypothesis on data (e.g. chi-squared test)

---

## Sample and Population

### Population:
- Amount of all items of relevance for an analysis.

### Sample
- Selection of items on basis of certain criteria (e.g. representativity) which
- will be analysed instead of the population

### Example opinion poll
- Population: all federal citizens who have a meaning
- Sample: the citizens who are polled by the polling organization

*complete record of all the values ↔ sampling*

**In archaeology only sampling is possible! The population can never be investigated!**

---

## Levels of measurement

- nominal:
  - Categories which do not have a defined relationship among each other, only counting is possible (e.g. sex)
- ordinal:
  - Categories which are comparable and differ from each other in their characteristic [size/power/intensity]; their rank is determinable (e.g. preservation conditions – bad, medium, good)
- metric:
  - Variable has a defined system of measurement, all calculations are possible. To distinguish are

1. interval: The variable has an arbitrary choosen neutral point (°C)
2. ratio: The variable has an absolute neutral point (°K)

- Sometimes also used: absolut scale
  - counts (number of inhabitans)

---

## Levels of measurement

![](data:image/png;base64,#../images/07_session/scales_of_measurements.png)

---

## Levels of measurement

<table class="table table-striped table-hover" style="margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:left;"> scale </th>
   <th style="text-align:left;"> Meaningful statements </th>
   <th style="text-align:left;"> Examples </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> nominal </td>
   <td style="text-align:left;"> equality, inequality </td>
   <td style="text-align:left;"> Telephon numbers, illnesses, ceramic types </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ordinal </td>
   <td style="text-align:left;"> bigger-smaller-relationship </td>
   <td style="text-align:left;"> Wind forces, academic ranks, classes of wealth, stratigraphic relations </td>
  </tr>
  <tr>
   <td style="text-align:left;"> interval </td>
   <td style="text-align:left;"> Equality of differences </td>
   <td style="text-align:left;"> Temperature in °C, calender age </td>
  </tr>
  <tr>
   <td style="text-align:left;"> ratio </td>
   <td style="text-align:left;"> Equality of ratios </td>
   <td style="text-align:left;"> Measurement of lengths, weight, height of a vessel </td>
  </tr>
</tbody>
</table>

.caption[after Bortz 2005]

---

## Inductive statistics or statistical inference

**Is used to draw conclusions about (unknown) parameters of the population on basis of a sample**
The results are always statistical ;-)

i.e. all statements are true with a certain probability but could be also false with a certain probability

The basis of statistical inference is probability theory (stochastic)

---
## Statistical hypothesis testing

### Validation of an assumption about the population

A assumption (hypothesis) about the population is made and than its probability is checked against the sample.

### Usual questions:

**How probable is it that two or more samples descend from the different/the same population?**

(eg. Is the custom of grave goods for man and women so different that two different social groups are visible?)

**How probable is it that a given sample descend from a population with certain parameters?**

(Is the amount of grave goods random or is a pattern visible?)

---

## Null hypothesis [1]

### Validation through falsification

In statistical tests most of the times not the statement is tested which one expects to be true but one tries to disprove the statement which one expects to be wrong: the null hypothesis.

This hypothesis states mostly, that a association do not exists or that there is no differences between the samples and the distribution of the observations is by chance.

Example: Is the composition of grave goods different between male and female deceased?

`\(H_0\)`: The compositionisthe same

`\(H_1\)`: The composition is different

### Reason
1. It is (logical) easier to prove, that a statement is wrong (falsify) then to prove that a statement is true (verify).
2. Most of the times it is easier to formulate a null hypothesis (How exactly is the composition different?). It doesn't make a assumption about how the character of a association/difference exactly is.

---

## Null hypothesis [2]

### „Workflow“ of a statistical test

**Construction of a alternative hypothesis:**

The composition of the grave goods is different between male and female deceased.

**Construction of the null hypothesis:**

The composition of the grave goods is the same in male and female burials.

**Test of the null hypothesis**

**If the result of the test is significant:**

Rejection of the null hypothesis, choice of the alternativ hypothesis. The composition of the grave goods is different between male and female deceased.
If the result of the test is not significant:

**The null hypothesis could not be rejected.**

We can not say if the composition of the grave goods is different between male and female deceased or not!

---

## Stat. Significance

### How true is true?

Statistical significance is effectively a measurement how probable a error is.

On basis of the significance the null hypothesis will be rejected and the alternative hypothesis will be choosen … or not.

There are classic boundary values for significance (significance levels):

0.05: significant, with 95% probability the decision is right.

0.01: very significant, with 99% probability the decision is right.

0.001: highly significant, with 99,9% probability the decision is right.

Often named with p-value or `\(\alpha\)`.

---

## Nonparametric tests

### Parametric vs. nonparametric

**Parametric**: The distribution of the values have to be in a certain form (e.g. normal distribution); assumptions about the distribution of the population are needed

**non-parametric**: no assumptions about the distribution of the sample and the population are needed

### Nonparametric tests, advantages and disadvantages:

**Advantage**: Also appropriate if no statements about the distribution are possible or the distribution fits no for parametric tests.

Also smaller samples are possible.

**Disadvantages**: Tests have general a lesser power.

---

## `\(\chi^2\)` test [1]

### Possible Questions

**Do settlements tend to be situated on rather good soil or is the distribution random?**

Conclusions about settlement behaviour and economy would be possible

**Do older individuals have more shoe-last celt as grave goods than younger?**

If shoe-last celt would be signs of social rank than this situation would make conclusions possible about heredity or acquisition of social rank during life time.

**Tests for nominal scaled variables are possible!**

Therefore of particular value for archaeology because we have often to deal with such data.

---

## Independent – dependent variable

### Independent Variable:
- The assumed cause of a relationship

### Dependent variable:
- The assumed effect of the independent variable in a relationship

### example:
- Number of pearls in a grave (Dependent) vs.
- sex of the deceased (independent)
- Hypothesis: The number of pearls in a grave depends on the sex of the deceased

### Can (have to be) not always to be defined
- e.g.: volume and height of a vessel...

---

## `\(\chi^2\)` test [2]

### Test for independence of two distributions

**Requirements**: at least 1 nominal scaled variable (one sample case) and
1 nominal scaled grouping variable (two sample case)

**Procedure with one sample**: observed values are compared with expected values given a certain distribution, no expected value should be < 5; n should be > 50

**Procedure with two samples**: observed values of both distributions are
compared with expected values if the samples would be even distributed,
no expected value should be < 5; n should be > 50

**Test statistics**: `\(\chi^2\)`

Significance depend on degree of freedom (df)

---

## Excursus degree of freedom