REPORTING INFERENTIAL STATISTICS IN APA FORMAT

This guide exists in order to provide guidance on reporting inferential results in psychology assignments. It does not provide guidance on scientific writing, nor does it aim to provide comprehensive information on how to structure practical reports or results sections, though some guidance is offered. In other words, this guide is aimed at ensuring that you report inferential statistics (i.e. the results of statistical analyses such as t-Tests, ANOVAs, and correlations) in the appropriate format.

SITE CONTENTS

1. Organising Results a) DESCRIPTIVES b) STATEMENT OF TEST c) FINDING INFERENTIALS d) IMPACT ON HYPOTHESIS 2. APA Inferentials a) General rules b) t-test results c) ANOVA/MANOVA results d) Correlation results e) Chi-square results f) Non-significant results g) Nearly’ significant results 3. Example SPSS output with inferentials a) t-test output b) ANOVA output c) Correlation output d) Chi-square output

4. Useful references

1.a - Descriptives At the beginning of your results section, you should present and explain the descriptive statistics that are relevant to your hypothesis. For example, if you have predicted that one group will score higher/lower than another group on a measure, it is vital to provide group means and standard deviations on that measure. Descriptive statistics can encompass measures of central tendency (e.g. means), measures of dispersion (e.g. standard deviations), and frequencies. Descriptive statistics are best presented in a table, such as the one below…

Table 1 – Mean weekly units of alcohol (with standard deviations) consumed by males and females

Group Mean Standard Deviation

MALE 180.30 40.04 FEMALE 8.40 1.20

Be sure to title and label adequately. Do not use raw output containing poor labels (e.g. SPSS sometimes produces tables with group codes, such as ‘1’ and ‘2’, where you’ve coded labels such as ‘male’ and ‘female’). Do not present tables and figures that are meaningless – tidy up the raw output to convey the full meaning. Do not simply put in the means/standard deviations output from SPSS. Make sure that you refer to the table in your results section. For example, “Table 1 shows that the mean weekly units of alcohol consumed by males was higher than that consumed by females.” If it would make your descriptive statistics more accessible to the reader, use a figure, such as a bar chart or line graph. Only provide such where they will enhance understanding of tables, and, again, make sure you refer to them in your results section. MAIN POINTS: -PRESENT DESCRIPTIVES FIRST

-TITLE & LABEL TABLES & FIGURES IN FULL -IF YOU INCLUDE TABLES & FIGURES, REFER TO THEM IN THE TEXT OF YOUR RESULTS SECTION -DO NOT INCLUDE INFORMATION FOR THE SAKE OF IT; AVOID NEEDLESS REPETITION

1.b - Statement of test

After providing your descriptive statistics, including any tables, figures, and explanations of them, you need to inform the reader as to the way in which the data was analysed to determine statistical significance. For example, if you used a t-Test to determine whether two groups had significantly different means, then you need to inform the reader of such. It is a good idea while stating the test used to offer some form of statistical justification for doing so (usually how your data meet the assumptions of the particular test you used). Be brief here unless you are using a less common statistical procedure or where your data may appear to infringe such assumptions. Then, you need to report the outcome of your statistical testing.

1.c - Finding inferentials

Before throwing any numbers at the reader, state in words the outcome of the test. If the test result shows that one group scored significantly higher, then say so… For example, “The t-test revealed that UCW graduates had significantly higher IQ scores than Cambridge graduates” BUT , directly after stating the outcome, you must provide the relevant inferential statistics. These are all available from your SPSS output (which you should include in an appendix to your report). A later section, called APA INFERENTIALS provides guidance on what format those inferentials are reported in, but for now, here is an example to demonstrate structure: “The t-Test revealed that UCW graduates had significantly higher IQ scores than Cambridge graduates, t (28) = 6.454, p < 0.05.” Notice how the inferentials follow immediately after the statement of the effect, separated by a comma. Here’s another example: “There was a significant positive correlation between age and fear of crime, revealing a tendency for more elderly individuals to be more fearful, r (50) = 0.76, p < 0.05.”

1.d - Impact on hypothesis

After stating the outcome of the analysis and the inferentials, it is important to state whether the result supports your hypothesis or not. For example, if our hypothesis was that there would be a positive relationship between income and stress, and our testing confirmed this, we might state: “The Pearson product-moment correlation test revealed a significant positive correlation between income and stress, r (240) = 0.56, p < 0.05. This supports hypothesis 1.” If we found the results did not fit our prediction, we might state: “The Pearson product-moment correlation test did not reveal a significant correlation between income and stress, r (240) = 0.12, p > 0.05. This does not support hypothesis 1, and the null cannot be rejected.” 2.a - APA Inferentials

GENERAL RULES…

APA format includes the following:

ALGABRAIC NAME OF STATISTIC (t, F, r, etc.)

DEGREES OF FREEDOM (if relevant)

VALUE OF THE STATISTIC

PROBABILITY LEVEL ASSOCIATED WITH THE STATISTIC (p<….)

Notice how the probability is reported in the various examples of significant findings. ‘p=’ is not used in APA format (though some other formats do so). ‘p <’ is used (i.e. ‘p is less than’). Therefore, you provide the alpha (or target p value) that your result is less than. Alpha values are conventionally accepted p levels that are regarded as the minimum for statistical significance – in psychology the least strict and most conventional alpha is 0.05. Where a p is less than 0.01, we say that the result is ‘highly significant’. If the p is less than the next alpha values (0.005, 0.001, 0.0005, 0.0001, etc…), it is not a good idea to use ever-more extreme terms to describe the significance (such as ‘devastatingly significant’ or ‘enormously significant’). Stick to ‘highly significant’ and you won’t get laughed at… For example, a p value of 0.032 is less than 0.05, but not less than 0.01. So we’d say p<0.05. If a p value was 0.009, it is less than 0.01, but not less than the next lowest

alpha of 0.005, so we’d say p<0.01. Get it?

2.b - t-Test results

Name, degrees of freedom, value, probability

t (??) = ????, p < ????

e.g. t (68) = 2.93, p < 0.05.

2.c - ANOVA/MANOVA results

Name, degrees of freedom, value, probability

F (??, ??) = ????, p < ????

e.g. F (2, 14) = 8.66, p < 0.05.

Notice that there are two values for degrees of freedom reported in the brackets after F. That is because with F ratios, there are two df values of relevance. The first is the df for the effect itself (see the df value in the row where the effect is stated in the SPSS output…). The second is the df for error (see the df value in the row where ERROR is stated in the SPSS output…). With one-way ANOVAS on SPSS, the output won’t provide a row named after your effect and a row named ERROR. Instead, it provides a BETWEEN row, and a WITHIN row. For the purposes of reporting such an F, df BETWEEN equals df for the effect, and df WITHIN equals df for ERROR. You should be able to think why this is the case from your knowledge of variance… 2.d – Correlation results

Name, degrees of freedom, value, probability

Name, number of cases, value*, probability

r (??) = ???? , p < ????

e.g. r (40) = 0.76, p<0.01

* if it is a negative r, say so! (i.e. –0.76)

Notice that in reporting a correlation, degrees of freedom aren’t included. Instead, N (number of participants) is provided. Degrees of freedom aren’t relevant to correlation analysis, but N is. Don’t forget to mention whether the correlation is positive or negative – this is vital. Also, state in words what tends to happen to one variable as the other changes. e.g. “As age increases, fear of crime tends to increase as well.” 2.e – Chi-Square results

Name, number of cases, value, probability

x2 (??) = ???? , p < ????

e.g. x2 (40) = 5.58, p<0.05

Note: Since chi-square deals with frequencies, means and standard deviations are inappropriate for the descriptives section of the report. What is more appropriate is a cross-tabulation table showing the expected and observed frequencies for the different levels of the categorical variables…

2.f – Results that do not reach statistical significance

State the outcome in words as usual, then provide all inferentials except instead of “p<”, use “p>0.05”.

e.g. t (48) = 1.02, p>0.05 2.g – ‘Nearly’ significant results…

Sometimes, the p is nearly significant, and you might want to point that out:

If so, the inferentials are identical other than

instead of saying “p<” you say “p=” and give the exact p value…

e.g. F (3, 46) = 5.44, p = 0.059

When stating the effect/difference/correlation in cases where the p is nearly significant , obviously you cannot say the result was significant. There are a few conventional ways of describing nearly significant results… e.g. 1 - While men did not score significantly higher than women on the OCI scale, the difference approached significance, t (24) = 3.22, p = 0.052. e.g. 2 - The positive correlation between anxiety and error score was not significant, though there was a trend toward significance, r (10) = 0.56, p = 0.054.

3.a - t-Test output example

The first thing to note is the significance of the Levene’s test for equality of variances. If the value is more than 0.05 (which it usually is…), we use the top row of the t-test output (equal variances assumed). If the value is less than 0.05, we use the bottom row of the t-test output (equal variances not assumed). The significance value from the Levene’s test in our output is less than 0.05, so we use the bottom row…

t (161) = 2.312, p < 0.05

Group Statistics

100 31.4000 20.3663 2.0366100 36.8800 12.1300 1.2130

SEXMaleFemale

Time, in minutes, spentin the bathroom

N MeanStd.

Deviation

Std.ErrorMean

Independent Samples Test

7.03 .009 -2.312 198 .022 -5.4800 2.3705 -10.15 -.8053

-2.312 161 .022 -5.4800 2.3705 -10.16 -.7988

EqualvariancesassumedEqualvariancesnotassumed

Time, inminutes, spentin thebathroom

F Sig.

Levene'sTest for

Equality ofVariances

t df

Sig.(2-tailed)

MeanDiffere

nce

Std.Error

Difference Lower Upper

95%Confidence

Interval of theDifference

t-test for Equality of Means

3.b – ANOVA output example

F (2, 27) = 12.45, p < 0.001 3.c – Correlation output

r (31) = 0.55, p < 0.001

ANOVA

5650.87 2 2825.4 12.46 .000

6120.10 27 226.67

11771.0 29

BetweenGroupsWithinGroupsTotal

NUMERICSCORE

Sum ofSquares df

MeanSquare F Sig.

Correlations

1.000 .555**

. .00131 31

.555** 1.000

.001 .31 31

PearsonCorrelationSig. (2-tailed)NPearsonCorrelationSig. (2-tailed)N

Exam performance(%)

Writing speed(words per minute)

Examperformance (%)

Writing speed(words per minute)

Correlation is significant at the 0.01 level (2-tailed).**.

3.d – Chi-square output

SEX * SMOKE Crosstabulation - Frequency developing exophasporal cancer

12 12 247.2 16.8 24.0

3 23 267.8 18.2 26.015 35 50

15.0 35.0 50.0

CountExpected CountCountExpected CountCountExpected Count

male

female

SEX

Total

smokers non-smokersSMOKE

Total

Chi-Square Tests

8.791b 1 .0037.055 1 .0089.219 1 .002

.005 .004

8.615 1 .003

50

Pearson Chi-SquareContinuity Correctiona

Likelihood RatioFisher's Exact TestLinear-by-LinearAssociationN of Valid Cases

Value df

Asymp.Sig.

(2-sided)Exact Sig.(2-sided)

Exact Sig.(1-sided)

Computed only for a 2x2 tablea.

0 cells (.0%) have expected count less than 5. The minimum expected count is7.20.

b.

x2 (50) = 8.791, p < 0.005

4. Useful References Sternberg, R. J. (1993). The Psychologist’s Companion. New York: Cambridge University Press. American Psychological Association (1987). Publication manual of the American Psychological Association (4th edition). Washington: APA. ALSO, VISIT

http://www.worc.ac.uk/departs/psycho/Teaching/guide1.html FOR GUIDANCE ON REFERENCING…

Comments or suggestions to s.ward@worc.ac.uk