BS900 Research Methods
BS900Research Methods
Introductory Guide to Statistics and SPSS (Part 1)
Revised September 2010
CONTENTS
Part 1:
Section 1 describing data
Section 2 significance
Section 3 non-parametric statistics
Section 4 parametric statistics
Part 2:
Section 5 correlations
Section 6 ANOVA
Section 7 multiple regression
Appendix
Saving and presenting data
Stats map
Worksheets
Excellent reference books are
Coolican, H. (2004) Research Methods and Statistics in Psychology.Hodder and Stoughton *
Howell, D.C. Statistical Methods for Psychology. Duxbury
Kinnear, P.R. and Gray (2007) SPSS 15 made Simple, Psychology Press, Taylor and Francis**
Tabachnick B.G. and FIDELL, L.S. (1996) Using Multivariate Statistics.Harper Collins.
*referred to as Coolican in the following text
** referred to as K&G in the following text
Section 1: Describing data
Data is a set of numbers. Before we start getting too excited about any data set or group or population it’s a good idea to be able to describe the sort of data we have.
Details of this are in the books (Coolican and K&G) and you should look them up.Types of data see Coolican p244-255.
We are primarily concerned about 1) LEVELS of data, and 2) DISTRIBUTION of data
The following are known as LEVELS
Nominal where information is put in to categories like sex ethnic group type of sport etc
Ordinalwhere cases are arranged in rank positions like 1st or 2nd.
Interval (Scale) where measurements are on a scale like height, weight, heart rate, etc, with equal distance between intervals
Ratio are interval data which has an absolute and necessary zero. So the time can be ratio (i.e. 0 seconds, whereas a temperature of zero doesn’t mean that there is no temperature)
Interval (scale) /ratio is the highest level of measurement, with nominal the lowest
So, Nominal data just put things in to a category (footballers, rugby players, gymnasts).
Ordinal tells you where you are in the ranks, e.g first in the race, last in the race.
Interval/ratio data is about measurement on a scale (because you know about this scale, you can say more sophisticated things about this data. You can do a lot more tests with this type of data). This makes it the highest level of measurement (highest= fanciest.).
Exercise 1: Collect the data
1)Measure and record the heart rate of everyone in the class
2)Create a list of all the heart rates
3)Organize the results from slowest to fastest and assign each person a rank
4)Put all the “male heart rates” in to one group and all the “female heart-rates” into
another.
Which is which type of data?
Ok so now we know what kind of data we have got. This is important because this determines what kind of statistics we’re able to perform on the data.
The next thing we might want to know is what to make of a particular individuals’ measurement i.e how does their heart–rate compare to other members of the population.
There are 3 ways to look at this;
Mean which is the average
Mode which is the most often occurring
Median the one in the middle when the data are in order
Exercise 2: Data distribution
The next thing we might like to know is something about how our data is distributed.
Normal Distribution (see Coolican 283 -288)
What most researchers, statisticians and (if they were animate) statistics want to see is a normally distributed population. This is based on the idea that in any population there will be lots of “typical” people (these are known as “normal” in mathematics and means just that i.e. a term used in mathematics ) and a few “atypical” ones.
If we think about height, most adults would be between 5 foot and 6 foot tall. These are the “typical” ones (normal in maths).
The further we go above or below these heights the fewer people there are and these are “atypical”.
In other words it is more unusual to be 7 foot or 4 foot tall, than say 5 foot 6 inches.
Because we want to be able to say most about most people (i.e. in this case the ones between 5 and 6 foot), we want to be sure that our population is “normally distributed”. That is, it has most people in the middle and a few either side.
Statistical analysis works best with these normally distributed populations.
Of course in sport or health this can be a bit of a worry since we are often dealing with abnormal or unusual populations (for example elite athletes or patients post MI)….. But we’ll think about that another day.
It is easier to see distributions if we plot them on a graph
A normal distribution looks like this…………………draw it here (Coolican p284)
A skewed distribution looks like this (a positive skew i.e. a disproportionate number of tall people)
Coolican p291 (draw it here)
or like this (a negative skew i.e. a disproportionate number of short people)Coolican p291 (draw it here)
If you have a skewed population and want to do statistical tests on the data you have a couple of different options…. Use a non parametric test, or adjust the data using an appropriate conversion (I will cover this another time)
Standard Deviation seeCoolican p 283 -287
This applies only to interval/ratio data. If you look this up in a statistics book (including Coolican) you will be faced with mathematics and will be told that a standard deviation is the square root of the variance. This is point when many people panic and run away. In practice the maths involved are not very scary but on this course you will not have to worry about them at all because we’ll let the computers do that! You do, though, need to understand what a standard deviation is, and why it is important.
Go to PASW 18.0 for windows
Click on type in data
Click on ok
Click on ‘VARAIBLE view’ tab at the bottom of the page
Click on variable name box and type in data_1
Click on ‘DATA view’tab at the bottom of the page
Click on the box with the yellow fill
Type in the first value, enter, continue until all data_1 values are entered
Go back to ‘VARIABLE view’ tab at the bottom of the page
Click on variable name box in row 2 and type in data_2
Click on ‘DATA view’tab at the bottom of the page
Input your data
Check your data is inputted correctly
Click on analyze /descriptive statistics/descriptive
Select both variables (data_1 and data_2_) and move them to the right hand box by clicking on the central arrow
Click on OK
Data 1 Data 2
6.007.00
3.009.00
6.005.00
5.004.00
9.007.00
5.001.00
1.005.00
4.009.00
7.004.00
3.003.00
5.001.00
6.008.00
5.002.00
6.002.00
4.008.00
In short a standard deviation tells you how far a score is away from the mean. It also gives you an idea of the dispersion or spread of the sample.
It can be shown mathematically that ;
within one standard deviation either side of the mean, 68.2% of all scores will be found,
within 2 standard deviations either side of the means, 95.44% of all scores will be found and
within 3 standard deviations either side of the mean 99.74% of all scores will be found.
So it is really useful thing to know because you can tell if (to return to our example) your heart rate is :-
Remarkable
Anything to be pleased about
Nothing to be pleased about
Something to worry about
Nothing to worry about
etc. etc.
Draw a normal distribution curve with standard deviations shown, and write in what percent of the population are contained within each SD.
Exercise 3: entering more complex data.
Next we are going to input our more data into the computer. What we hope to achieve here to get the computer to work out your mean median mode and the standard deviation of your data, according to group.
Go to PASW 18.0 for windows
Click on type in data
Click on ok
Click on VARIABLE view
Click on variable name box and type in hrt_rate
Click on DATA view
Click on the box with the yellow fill
Type in the first value, enter, Type in the 2nd value…and so on until you have typed in all the data
Check your data is inputted correctly
In the same way, in the ‘VARIABLE view’ window, create a column named ‘gender’
-You can label these columns M and F if you change the ‘type’ to ‘string’ or you can label them 1 and 2 and create labels or ‘values’ that will appear on your output. This really helps if you have a lot of columns all labelled 1 or 2 etc
Click on analyze /descriptive statistics/ frequencies*
Click on the central arrow (this moves hrt_rate to the variables box)(if you have input more than one variable you will need to select the variable (s) you wish to analyse )
Click on statistics
Click on mean, mode, median, standard deviation
Click on continue
Click on ok
See what you get!
- *Note you can do broadly similar basic activities using the ‘frequencies’ or the ‘descriptive’ icons. For more complex data analysis use the ‘frequencies’ option.
To look at the mean etc ACCORDING TO GROUP, you need to
Click on Analyze/ Descriptives /Explore
In this scenario, ‘hrt_rate’ is the dependent variable and ‘gender’ is the factor.
See what you get!
For the standard deviation (SD)
thebigger the number (in relation to the size of the population) the more variability in the data, i.e. the wider the distribution of the scores
thelower the number, the less variability in the data, i.e. the narrower the variability scores
You can also again information about the shape of the distribution by the kurtosis (shape of the curves) and the skewness. Check the boxes for these components when you are in the ‘Frequencies’ box.
Kurtosis refers to the ‘peakness of the distrubution’ whilst skew refers to the ‘asymetry’ of the distribution
Section 2: Significance
(Please refer to Coolican chapter 11 (p313 on)
This concept is at the centre of statistical analysis. Let us take a typical sports science experiment.
We think (hypothesise) that our new training programme benefits fitness
We take 2 groups and measure their fitness
Group 1 continues to pursue its ‘regular’ fitness programme
Group 2 follows our ‘new’ fitness programme
3 months later we measure fitness again
We want to know if the 2 groups are different (i.e. does our new programme work/ does our new programme improve fitness more than the old programme?) Clearly there will be all sorts of other variables sloshing around but lets just stick with the numbers.
We can see that the individual and average fitness scores have changed but:-
Are these changes meaningful?
Should we take any notice of them?
In other words
Are these differences statistically significant?
Being able to see that there is a difference is nothing like the same as saying that the differences between them are statistically significant. NB this is not the same as saying a change is clinically significant!
We get at this question of significance by considering the notion of probability.
We end up saying that these differences are probably significant. The good news is that we can say how probable it is that these differences are significant.
Often we can very confident that we are correct.
You might find it interesting to note that statistics came from the law courts of 19th century France where mathematicians were asked to say how likely it was that a given casino gambling game was fair. The critical question was to resolve was the part played by chance.
When we perform statistical analysis today we are trying to do the same thing. That is, decide whether our results are due to our experimental manipulationOR are due to chance. When you do anything with science (especially if people are involved) it is always as well to watch out for chance!
Obviously there are all sorts of issues about which statistical test to use.
This is about choosing the correct tool for the job (e.g. there is little point using a lawn mower to clean your car) but first we need to step back and think about the concepts involved in these ideas about significance.
In order to be sure if our findings are statistically significant first we have to know what it is that we are trying to find out. In other words we need a hypothesis – we state what we think will happen as a result of our experimental intervention.
For our training experiment the hypothesis was
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
We also need a NULL hypothesis which states the opposite of the experimental hypothesis.The null hypothesis states that any difference between the 2 groups is due to chance (i.e. nothing we did in the experiment will have any effect on fitness)
For our training experiment the null hypothesis was
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
It is the NULL hypothesis that we accept or reject according to our statistical test.
Accepting the NULL hypothesis means that to some extent and/or at some point we were wrong, i.e. our plan to increase fitness using our new training programme was not successful. There was NO EFFECT on fitness due to our intervention
Rejecting the NULL hypothesis (and accept the experimental hypothesis) then we were correct i.e. our new training programme was more successful.
We accept or reject these hypotheses on the basis of significance which in turn is related to probability.
There is a convention about this which puts significance at better than or equal to 5% (or 0.05). In other words we accept something as being statistically significant if we think it will happen 95 times out of 100. We leave the other 5 times to chance.
In the case of things like drug trials or invasive surgery we ask for a more stringent level of significance often 1% since we cannot afford to be killing or harming people!
Exercise:
Remember, the significance level tells us whether to accept or reject the null hypothesis.
If the level is ABOVE 0.05 we ACCEPT the null hypothesis (which usually implies therefore that we have to accept the fact that our experiment or intervention did not work, or that there was no statistical difference between our groups).
If the level is BELOW 0.05, we reject the Null hypothesis and can generally conclude that the groups are different and our experiment was successful or our groups were different and so on..
SO: imagine you have conducted some statistical tests (it does not matter at the moment which tests you would do)
Given these p values shown, decide what you would do with your null hypothesis and what your conclusion would be regarding your experiment.
1.You conduct a test to determine whether men are taller that women. Think about what your null hypothesis and your experimental hypothesis would be. Your statistical test gives you a p value of 0.03.
2.You conduct a study to see if fast walking has a different effect on weight loss that slow walking, over the same distance. Statistical tests provide a p value of 0.08
- You conduct a test to see if heart rate is lower in the morning than in the afternoon. Your statistical test gives a p value of 0.56
- You conduct a test to see if patients have higher self esteem at the end of a 12 week exercise programme compared with before the programme. Your statistical test gives a p value of 0.048
Type 1 and Type 2 errors
These kinds of errors are common but should be avoided
Type 1 (Type I)
Where we reject the Null hypothesis when we should accept it. In other words say we have a statistically significant result when we do not. There are lots of ways of making this mistake e.g. setting our significance level too low, performing the wrongs stats, having a flawed methodology wanting to be correct too badly etc. etc..
Type 2 (Type II)
Where we accept the Null hypothesis when we should have rejected it. In other words say we do not have a statistically significant result when we do. In other, other words say we are wrong when we are correct (dumb or what!). Again this is easy to do but it is understandable that a researcher would want to be cautious.
The Null Hypothesis is…..What we do / True / False
Accept
Reject
The last piece of this jigsaw is the idea of One- and Two-tailed tests.
It is VERY important to get this right, as getting it wrong makes it easy to make a type 1 or a type 2 error.
If we return to our training example, we hoped that our new training program would improve the fitness of our players.
If our hypothesis was
1)“Our new training program will (statistically) significantly improve fitness”
then because we are making a prediction about the type of change we expect (i.e. an improvement) then we call this a one tail hypothesis. When we do this we say that we are predicting a direction, in this case the direction is, an improvement. If we were predicting a decrement in performance then I hope you can see that this is also a 1 tailed hypothesis because we are still predicting a direction.
If our hypothesis was
2)“Our new training program will (statistically) significantly change fitness”
then because we are making a prediction that a change will occur but we are not sure whether it will be an improvement or a decrement then we call this a two tail hypothesis. When we do this we say that we are not predicting a direction, we are just predicting a change.
If you were wondering where the tail bit comes in look at these graphs.
This is a One tail hypothesis graph. It shows you that we are interested in just one end of the graph. We call this end a tail.
This is a Two tail hypothesis graph. It shows you that we are interested in both ends of the graph. Again we call these end tails.
Both of these graphs refer to the theoretical distribution of a statistic. As you can see the distribution has 2 tails and a central hump. In order to reject the null hypothesis we must have some value which is so unusual that it is unlikely that it is due to chance. Thus we only consider the tails of our distribution, as the more common values occur in the centre (remember?).
We also know that only values that can occur by chance only 5% or less of the time are extreme enough to be deemed significant. That is extreme enough to let us reject the null hypothesis. So when we consider a one or a two-tailed hypothesis we are only considering 5% of the distribution.
A two-tailed hypothesis is in effect saying that “the effect of the independent variable will be to produce a value which is extreme, but which could be located at either end of the distribution.