How to reduce many survey questions to a few summary scores

01What is principal component analysis?

Imagine you asked staff nine questions about their job. Many of the answers go together: people who say they have too much work also tend to say their deadlines are tight. Principal component analysis, or PCA, uses that overlap. Instead of working with nine separate answers, it builds a few summary scores that capture most of what the nine answers say. Each summary score is called a component.

The first component captures the biggest pattern in the answers. The second captures the next biggest pattern that is separate from the first, and so on. Usually the first few components hold most of the information, so you can describe the data with far fewer numbers.

An example

You survey 320 staff members about their working conditions. There are nine statements, each rated from 1 (strongly disagree) to 5 (strongly agree). Four are about workload: too much work, tight deadlines, unpaid overtime and feeling stressed. Four are about support from management: the manager listens, feedback is useful, I feel recognised, and I trust the leadership. The ninth says that pay is fair. You want to know whether nine items can be summarised by two or three scores, which items travel together, and whether any item adds little. PCA answers all three.

What you get out of it

1. Components
   The summary scores, from the one that holds the most information to the one that holds the least.
2. Variance explained
   How much of the information in your variables each component holds, and how much the ones you keep hold together.
3. Loadings
   How closely each question is tied to each component. A higher number means a closer tie.
4. Communalities
   How well the kept components represent each question. A low value means the question is left out of the picture.

02When to use it

Use PCA when you have several numeric variables that are correlated and you want to simplify them. Common reasons are to describe the main patterns in a set of questionnaire items, to see which items overlap before building a scale, or to replace many overlapping variables with a few summaries for a later analysis such as linear regression. The variables can be continuous, or ratings such as 1 to 5 that you are willing to treat as numbers. Categories without an order, like district or occupation, do not belong in PCA. The module needs at least three variables.

PCA will not help when the variables are barely correlated, because there is then nothing to condense. The KMO value, described below, checks this. To test whether the items of a scale are consistent, use Cronbach's alpha. To look at the relationship between just two variables, use correlation.

03Components, eigenvalues and loadings

Components

Picture each respondent as a point in a space with one axis per variable. The cloud of points is longer in some directions than in others. The first component is the line through the cloud along its longest direction. The second is the longest direction at right angles to the first, and so on. Each respondent gets a position along each of these new lines.

Eigenvalues

When the variables are standardised, each one has a variance of 1, so the total variance equals the number of variables. The eigenvalue of a component says how much of that total it carries, in units of one variable. An eigenvalue of 3 means the component carries as much variation as three of the original variables together. An eigenvalue below 1 means the component carries less than a single original variable, which is a poor return for a summary.

Loadings

A loading is the link between a variable and a component. When the variables are standardised, it is the correlation between the two, so it runs from -1 to 1. Squaring a loading gives the share of that variable's variation that the component explains. The sign tells you the direction, and variables with the same sign on a component move together.

Communality

The communality of a variable is the sum of its squared loadings across the retained components. It is the share of the variable's variation that the retained components capture. What is left over, 1 minus the communality, is lost when you drop the other components.

04How PCA is calculated

1. Standardise
   Each variable is rescaled to a mean of 0 and a standard deviation of 1, so that no variable dominates because it has a larger range. The module does this by default with Standardise variables. Without it, the calculation uses covariances, and a variable with a larger spread has more influence.
2. Build the correlation matrix
   Work out the correlation between every pair of variables.
3. Find the eigenvalues and eigenvectors
   A standard matrix calculation splits the matrix into components. The eigenvalues are sorted from largest to smallest.
4. Scale the eigenvectors
   Multiply each eigenvector by the square root of its eigenvalue to get the loadings.
5. Add up
   Square and sum the loadings for communalities, and divide each eigenvalue by the total to get the percentage of variance.

In the staff survey there are nine standardised variables, so the total variance is 9. The first eigenvalue is 2.988, which gives 2.988 / 9 = 33.2% of the variance. The variable Tight deadlines has loadings of 0.683 and -0.415 on the first two components, so its communality is 0.683² + 0.415² = 0.639.

Checking that PCA makes sense: KMO

The Kaiser-Meyer-Olkin measure, or KMO, compares the ordinary correlations between variables with their partial correlations, which are the correlations left once the other variables are taken into account. If variables share a lot, the ordinary correlations are large and the partial ones are small, and KMO is close to 1. If the partial correlations are as large as the ordinary ones, there is little shared pattern to summarise.

All pairs of different variables are included. The module labels the result as follows, and the help text suggests that 0.60 is the lowest value to accept.

Choosing how many components to keep

There is no single correct answer. Three guides are in common use, and they are best used together.

• Kaiser rule. Keep components with an eigenvalue of 1 or more. The module marks these in the variance table. The rule tends to keep too many components when there are many variables.
• Scree plot. Plot the eigenvalues in order and look for the bend, or elbow, where the line stops dropping steeply. Keep the components before the bend.
• Variance explained. Many analysts like the retained components to carry 60% to 70% of the variance. The module calls 70% or more strong and 50% or more moderate.

Interpretability counts as well. A component you cannot describe in words is a weak summary, even if it passes the other tests. In the module you type in the number of components to extract, and it tells you how many the Kaiser rule would keep if that differs from your choice.

05Reading the output

At the top of the results you see the number of components extracted, the percentage of variance they explain together, and the KMO value with its label. Three tabs follow.

Variance explained

A table lists every component with its eigenvalue, percentage of variance, cumulative percentage and a mark where the Kaiser rule says to keep it. Under it is a scree plot. Retained components are drawn in green, the others in grey, and a dashed line marks an eigenvalue of 1.

Component loadings

A heatmap shows the loading of each variable on each component, shaded by size: strong at 0.60 or more, moderate at 0.40 or more, weak at 0.30 or more, and grey below that. Two display options decide which variables count as belonging to a component.

• Dominant factor places each variable under the one component where its loading is largest, provided that loading reaches the minimum.
• By threshold places a variable under every component where its loading reaches the minimum, so one variable can appear more than once.

The minimum loading is 0.35 to start with and can be changed between 0.20 and 0.80. These settings also decide how the variables are grouped in the table that you can send to a Data Canvas story.

Communalities

Each variable's communality is listed. Values below 0.40 are highlighted and named under the table, because a variable with a low communality is not well represented by the retained components.

The interpretation text

The module writes a summary that states how suitable the data are, how many components were extracted and how much variance they keep, which variables define each component, and which variables have low communality.

06How to run it in AnalyZ Solutions

Open Scale & Reliability, then choose Principal Component Analysis.

1. Select the variables
   Choose three or more numeric variables, including rating items. Leave out identifiers and variables that are not meant to be summarised.
2. Set the number of components
   The starting value is 3, and it cannot be larger than the number of variables. Because the number is chosen before the run, run PCA once with a generous number, read the eigenvalues and the scree plot, then run it again with the number you settle on.
3. Keep Standardise variables ticked
   It is recommended, and necessary when the variables are on different scales.
4. Apply a subsample filter if needed
   Use it to run the analysis for one group only.
5. Click Run PCA
   You get the summary figures, the three tabs and the written interpretation. Export the tables as CSV, Excel or Word, or add the result to AnalyZense or to a Data Canvas story.

People with a missing value on any selected variable are left out, and the number of cases used is shown above the results. When you change the number of components, the loadings on the first components stay the same. Communalities and the cumulative variance change, because they depend on how many components are kept.

07Interpreting the result

The figures below are invented to show the steps, and they follow the results tables in AnalyZ Solutions, with some rows left out to save space. They come from the staff survey described earlier: 320 people, nine items rated from 1 to 5, two components extracted with standardised variables.

Reading it step by step

1. Check the suitability
   KMO is 0.807, labelled meritorious. The items share enough for PCA to be worthwhile.
2. Decide on the number of components
   Two eigenvalues are above 1 (2.99 and 2.11), the third is 0.83, and the scree plot bends after the second. Two components carry 56.6% of the variance, which the module calls moderate. The Kaiser count matches the number extracted, so no warning appears.
3. Read component 1
   All nine items have positive loadings, from 0.49 to 0.68. This is the usual pattern for a first component. It captures what the items have in common, which here is partly a general tendency to agree with the statements. It does not separate the two topics.
4. Read component 2
   The workload items have negative loadings (-0.53 to -0.41) and the support items have positive ones (0.52 to 0.63). The pay item is close to zero. Component 2 therefore contrasts workload with support, and this is where the two topics show up.
5. Look at the communalities
   Eight items are between 0.53 and 0.64. Pay is fair has 0.243, below 0.40, and is highlighted. It has a moderate loading on component 1 and none on component 2, so the two components capture little of what is specific to that item.
6. Treat the grouping with care
   In the Dominant factor view with a minimum of 0.35, Feel recognised goes under component 1 (0.54 against 0.52) and Trust leadership under component 2 (0.52 against 0.51), although both are support items. The gaps are only 0.02 and 0.02. When an item sits this close to the dividing line, an unrotated solution cannot place it reliably. A factor analysis with rotation would.

In plain words

Two components summarise the nine staff survey items, carrying 57% of their variation. The first reflects what all the items share. The second separates the workload items from the support items. Pay is fair is poorly captured by either. Because the components are unrotated, exploratory factor analysis is the better step for deciding which items belong together.

08Common mistakes

Treating components as proven constructs. PCA describes patterns in the data. It does not show that hidden traits exist.

Keeping components by the Kaiser rule alone. Look at the scree plot and the meaning of each component too.

Skipping standardisation. Without it, a variable measured on a wide scale can dominate the first component for no substantive reason.

Reading the sign of a component literally. The sign of a whole component can flip without changing anything of substance. Compare signs across variables within a component.

Including variables that do not belong. A variable with a low communality and weak loadings adds noise. Check KMO and the communalities, then consider removing it.

Too few cases. A common guide is at least five respondents per variable, and more is better. With few cases the components are unstable.

Using unordered categories. Codes for district or occupation are labels, not quantities, and should not be entered.