Normalizing Variable Transformations – 6 Simple Options

• Overview Transformations
• Normalizing - What and Why?
• Normalizing Negative/Zero Values
• Test Data
• Descriptives after Transformations
• Conclusion

Overview Transformations

Normalizing - What and Why?

“Normalizing” means transforming a variable in order to
make it more normally distributed. Many statistical procedures require a normality assumption: variables must be normally distributed in some population. Some options for evaluating if this holds are

• inspecting histograms;
• inspecting if skewness and (excess) kurtosis are close to zero;
• running a Shapiro-Wilk test and/or a Kolmogorov-Smirnov test.

For reasonable sample sizes (say, N ≥ 25), violating the normality assumption is often no problem: due to the central limit theorem, many statistical tests still produce accurate results in this case. So why would you normalize any variables in the first place? First off, some statistics -notably means, standard deviations and correlations- have been argued to be technically correct but still somewhat misleading for highly non-normal variables.

Second, we also encounter normalizing transformations in multiple regression analysis for

• meeting the assumption of normally distributed regression residuals;
• reducing heteroscedasticity and;
• reducing curvilinearity between the dependent variable and one or many predictors.

Normalizing Negative/Zero Values

Some transformations in our overview don't apply to negative and/or zero values. If such values are present in your data, you've 2 main options:

1. transform only non-negative and/or non-zero values;
2. add a constant to all values such that their minimum is 0, 1 or some other positive value.

The first option may result in many missing data points and may thus seriously bias your results. Alternatively, adding a constant that adjusts a variable's minimum to 1 is done with

$$Pos_x = Var_x - Min_x + 1$$

This may look like a nice solution but keep in mind that a minimum of 1 is completely arbitrary: you could just as well choose, 0.1, 10, 25 or any other positive number. And that's a real problem because the constant you choose may affect the shape of a variable's distribution after some normalizing transformation. This inevitably induces some arbitrariness into the normalized variables that you'll eventually analyze and report.

Test Data

We tried out all transformations in our overview on 2 variables with N = 1,000:

• var01 has strong negative skewness and runs from -1,000 to +1,000;
• var02 has strong positive skewness and also runs from -1,000 to +1,000;

These data are available from this Googlesheet (read-only), partly shown below.

SPSS users may download the exact same data as normalizing-transformations.sav.

Since some transformations don't apply to negative and/or zero values, we “positified” both variables: we added a constant to them such that their minima were both 1, resulting in pos01 and pos02.

Although some transformations could be applied to the original variables, the “normalizing” effects looked very disappointing. We therefore decided to limit this discussion to only our positified variables.

Square/Cube Root Transformation

A cube root transformation did a great job in normalizing our positively skewed variable as shown below.

The scatterplot below shows the original versus transformed values.

SPSS refuses to compute cube roots for negative numbers. The syntax below, however, includes a simple workaround for this problem.

Logarithmic Transformation

A base 10 logarithmic transformation did a decent job in normalizing var02 but not var01. Some results are shown below.

The scatterplot below visualizes the original versus transformed values.

SPSS users can replicate these results from the syntax below.

Power Transformation

A third power (or cube) transformation was one of the few transformations that had some normalizing effect on our left skewed variable as shown below.

The scatterplot below shows the original versus transformed values for this transformation.

These results can be replicated from the syntax below.

Inverse Transformation

The inverse transformation did a disastrous job with regard to normalizing both variables. The histograms below visualize the distribution for pos02 before and after the transformation.

The scatterplot below shows the original versus transformed values. The reason for the extreme pattern in this chart is setting an arbitrary minimum of 1 for this variable as discussed under normalizing negative values.

SPSS users can use the syntax below for replicating these results.

Hyperbolic Arcsine Transformation

As shown below, the hyperbolic arcsine transformation had a reasonably normalizing effect on var02 but not var01.

The scatterplot below plots the original versus transformed values.

In Excel and Googlesheets, the hyperbolic arcsine is computed from =ASINH(...) There's no such function in SPSS but a simple workaround is using

$$Asinh_x = ln(Var_x + \sqrt{Var_x^2 + 1})$$

The syntax below does just that.

Arcsine Transformation

Before applying the arcsine transformation, we first rescaled both variables to a range of [-1, +1]. After doing so, the arcsine transformation had a slightly normalizing effect on both variables. The figure below shows the result for var01.

The original versus transformed values are visualized in the scatterplot below.

The rescaling of both variables as well as the actual transformation were done with the SPSS syntax below.

Descriptives after Transformations

The figure below summarizes some basic descriptive statistics for our original variables before and after all transformations. The entire table is available from this Googlesheet (read-only).

Conclusion

If we only judge by the skewness and kurtosis after each transformation, then for our 2 test variables

• the third power transformation had the strongest normalizing effect on our left skewed variable and
• the cube root transformation worked best for our right skewed variable.

I should add that none of the transformations did a really good job for our left skewed variable, though.

Obviously, these conclusions are based on only 2 test variables. To what extent they generalize to a wider range of data is unclear. But at least we've some idea now.

If you've any remarks or suggestions, please throw us a quick comment below.

Thanks for reading.

How to Find & Exclude Outliers in SPSS?

• Method I - Histograms
• Excluding Outliers from Data
• Method II - Boxplots
• Method III - Z-Scores (with Reporting)
• Method III - Z-Scores (without Reporting)

Summary

Outliers are basically values that fall outside of a normal range for some variable. But what's a “normal range”? This is subjective and may depend on substantive knowledge and prior research. Alternatively, there's some rules of thumb as well. These are less subjective but don't always result in better decisions as we're about to see.

In any case: we usually want to exclude outliers from data analysis. So how to do so in SPSS? We'll walk you through 3 methods, using life-choices.sav, partly shown below.

In this tutorial, we'll find outliers for these reaction time variables.

During this tutorial, we'll focus exclusively on reac01 to reac05, the reaction times in milliseconds for 5 choice trials offered to the respondents.

Method I - Histograms

Let's first try to identify outliers by running some quick histograms over our 5 reaction time variables. Doing so from SPSS’ menu is discussed in Creating Histograms in SPSS. A faster option, though, is running the syntax below.

Result

Let's take a good look at the first of our 5 histograms shown below.

The “normal range” for this variable seems to run from 500 through 1500 ms. It seems that 3 scores lie outside this range. So are these outliers? Honestly, different analysts will make different decisions here. Personally, I'd settle for only excluding the score ≥ 2000 ms. So what's the right way to do so? And what about the other variables?

Excluding Outliers from Data

The right way to exclude outliers from data analysis is to specify them as user missing values. So for reaction time 1 (reac01), running missing values reac01 (2000 thru hi). excludes reaction times of 2000 ms and higher from all data analyses and editing. So what about the other 4 variables?

The histograms for reac02 and reac03 don't show any outliers.

For reac04, we see some low outliers as well as a high outlier. We can find which values these are in the bottom and top of its frequency distribution as shown below.

If we see any outliers in a histogram, we may look up the exact values in the corresponding frequency table.

We can exclude all of these outliers in one go by running missing values reac04 (lo thru 400,2085). By the way: “lo thru 400” means the lowest value in this variable (its minimum) through 400 ms.

For reac05, we see several low and high outliers. The obvious thing to do seems to run something like missing values reac05 (lo thru 400,2000 thru hi). But sadly, this only triggers the following error:

The problem here is that you can't specify a low and a high
range of missing values in SPSS. Since this is what you typically need to do, this is one of the biggest stupidities still found in SPSS today. A workaround for this problem is to

• RECODE the entire low range into some huge value such as 999999999;
• add the original values to a value label for this value;
• specify only a high range of missing values that includes 999999999.

The syntax below does just that and reruns our histograms to check if all outliers have indeed been correctly excluded.

Result

First off, note that none of our 5 histograms show any outliers anymore; they're now excluded from all data analysis and editing. Also note the bottom of the frequency table for reac05 shown below.

Low outliers after recoding and labelling are listed under Missing.

Even though we had to recode some values, we can still report precisely which outliers we excluded for this variable due to our value label.

Before proceeding to boxplots, I'd like to mention 2 worst practices for excluding outliers:

• removing outliers by changing them into system missing values. After doing so, we no longer know which outliers we excluded. Also, we're clueless why values are system missing as they don't have any value labels.
• removing entire cases -often respondents- because they have 1(+) outliers. Such cases typically have mostly “normal” data values that we can use just fine for analyzing other (sets of) variables.

Sadly, supervisors sometimes force their students to take this road anyway. If so, SELECT IF permanently removes entire cases from your data.

Method II - Boxplots

If you ran the previous examples, you need to close and reopen life-choices.sav before proceeding with our second method.

We'll create a boxplot as discussed in Creating Boxplots in SPSS - Quick Guide: we first navigate to Analyze Descriptive Statistics Explore as shown below.

Next, we'll fill in the dialogs as shown below.

Completing these steps results in the syntax below. Let's run it.

Result

Quick note: if you're not sure about interpreting boxplots, read up on Boxplots - Beginners Tutorial first.

Our boxplot indicates some potential outliers for all 5 variables. But let's just ignore these and exclude only the extreme values that are observed for reac01, reac04 and reac05.

So, precisely which values should we exclude? We find them in the Extreme Values table. I like to copy-paste this into Excel. Now we can easily boldface all values that are extreme values according to our boxplot.

Copy-pasting the Extreme Values table into Excel allows you to easily boldface the exact outliers that we'll exclude.

Finally, we set these extreme values as user missing values with the syntax below. For a step-by-step explanation of this routine, look up Excluding Outliers from Data.

Method III - Z-Scores (with Reporting)

A common approach to excluding outliers is to look up which values correspond to high z-scores. Again, there's different rules of thumb which z-scores should be considered outliers. Today, we settle for |z| ≥ 3.29 indicates an outlier. The basic idea here is that if a variable is perfectly normally distributed, then only 0.1% of its values will fall outside this range.

So what's the best way to do this in SPSS? Well, the first 2 steps are super simple:

• we add z-scores for all relevant variables to our data and
• see if their minima or maxima meet |z| ≥ 3.29.

Funnily, both steps are best done with a simple DESCRIPTIVES command as shown below.

Result

Minima and maxima for our newly computed z-scores.

Basic conclusions from this table are that

• reac01 has at least 1 high outlier;
• reac02 and reac03 don't have any outliers;
• reac04 and reac05 both have at least 1 low and 1 high outlier.

But which original values correspond to these high absolute z-scores? For each variable, we can run 2 simple steps:

• FILTER away cases having |z| < 3.29 (all non outliers);
• run a frequency table -now containing only outliers- on the original variable.

The syntax below does just that but uses TEMPORARY and SELECT IF for filtering out non outliers.

Result

Finding outliers by filtering out all non outliers based on their z-scores.

Note that each frequency table only contains a handful of outliers for which |z| ≥ 3.29. We'll now exclude these values from all data analyses and editing with the syntax below. For a detailed explanation of these steps, see Excluding Outliers from Data.

Method III - Z-Scores (without Reporting)

We can greatly speed up the z-score approach we just discussed but this comes at a price: we won't be able to report precisely which outliers we excluded. If that's ok with you, the syntax below almost fully automates the job.

Result

The frequency table below tells us that 4 outliers having |z| ≥ 3.29 were excluded for reac04.

Under Missing we see the number of excluded outliers but not the exact values.

Sadly, we're no longer able to tell precisely which original values these correspond to.

Final Notes

Thus far, I deliberately avoided the discussion precisely which values should be considered outliers for our data. I feel that simply making a decision and being fully explicit about it is more constructive than endless debate.

I therefore blindly followed some rules of thumb for the boxplot and z-score approaches. As I warned earlier, these don't always result in good decisions: for the data at hand, reaction times below some 500 ms can't be taken seriously. However, the rules of thumb don't always exclude these.

As for most of data analysis, using common sense is usually a better idea...

Thanks for reading!

How to Convert String Variables into Numeric Ones?

Introduction & Practice Data File

Converting an SPSS string variable into a numeric one is simple. However, there's a huge pitfall that few people are aware of: string values that can't be converted into numbers result in system missing values without SPSS throwing any error or warning.

This can mess up your data without you being aware of it. Don't believe me? I'll demonstrate the problem -and the solution- on convert-strings.sav, part of which is shown below.

SPSS Strings to Numeric - Wrong Way

First off, you can convert a string into a numeric variable in variable view as shown below.

Now, I never use this method myself because

• I can't apply it to many variables at once, so it may take way more effort than necessary;
• it doesn't generate any syntax: there's no Paste button and nothing's appended to my journal file;
• it can mess up the data. However, there's remedies for that.

So What's the Problem?

Well, let's do it rather than read about it. We'll

1. set empty cells as user missing values for s3;
2. convert s3 to numeric in variable view;
3. run descriptives on the result.

Result

Note that some values in our string variable have been flagged with “a”. We probably want these to be converted into numbers. We have 459 valid values (non empty cells).

After converting our variable to numeric, we ran some descriptives. Note that we only have N = 444. Apparently, 15 values failed to convert -probably not what we want. And we usually won't notice this problem because we don't get any warning or error.

Conversion Failures - Simplest Solution

Right, so how can we perform the conversion safely? Well, we just

• inspected frequency tables: how many non empty values do we have before the conversion?
• converted our variable(s) to numeric;
• inspected N in a descriptive statistics after the conversion. If N is lower than the number of non empty string values (frequencies before conversion), then something may be wrong.

In our first example, the frequency table already suggested we must remove the “a” from all values before converting the variable. We'll do just that in a minute.
Although safe, I still think this method is too much work, especially for multiple variables. Let's speed things up by using some handy syntax.

SPSS - String to Numeric with Syntax

The fastest way to convert string variables into numeric ones is with the ALTER TYPE command.This requires SPSS version 16 or over. For SPSS 15 or below, use the NUMBER function. It allows us to convert many variables with a single line of syntax.
The syntax below converts all string variables in one go. We then check a descriptives table. If we don't have any system missing values, we're done.

SPSS ALTER TYPE Example

Note: using alter type s1 to s4 (f1). will also work but the decimal places for s4 won't be visible. This is why we set the correct f format: f6.3 means 6 characters including the decimal separator and 3 decimal places as in 12.345. Which is the format of our string values.

Result

Since we've 480 cases in our data, we're done for s1. However, the other 3 variables contain system missings so we need to find out why. Since we can't undo the operation, let's close our data without saving and reopen it.

Solution 2: Copy String Variables Before Conversion

Things now become a bit more technical. However, readers who struggle their way through will learn a very efficient solution that works for many other situations too. We'll basically

• copy all string variables;
• convert all string variables;
• compare the original to the converted variables.

Precisely, we'll flag non empty string values that are system missing after the conversion. As these are at least suspicious, we'll call those conversion failures. This may sound daunting but it's perfectly doable if we use the right combination of commands. Those are mainly STRING, RECODE, DO REPEAT and IF.

Copy and Convert Several String Variables

Result

Only flag3 and flag4 contain some conversion failures. We can visually inspect what's the problem by moving these cases to the top of our dataset.

Result

Remove Illegal Characters, Copy and Convert

Some values are flagged with letters “a” through “e”, which is why they fail to convert. We'll now fix the problem. First, we close our data without saving and reopen it. We then rerun our previous syntax but remove these letters before the conversion.

Syntax

Result

All flag variables contain only (system) missings. This means that we no longer have any conversion failures; all variables have been correctly converted. We can now delete all copy and flag variables, save our data and move on.

Thanks for reading!

SPSS Variable Types and Formats

Understanding SPSS variable types and formats allows you to get things done fast and reliably. Getting a grip on types and formats is not hard if you ignore the very confusing information under variable view. This tutorial will put you on the right track.
We encourage you to follow along with this tutorial by downloading and opening computer_parts.sav, partly shown below.

SPSS Variable Types

SPSS has 2 variable types:

• Numeric variables contain only numbers and are suitable for numeric calculations such as addition and multiplication.
• String variables may contain letters, numbers and other characters. You can't do calculations on string variables -even if they contain only numbers.

There are no other variable types in SPSS than string and numeric. However, numeric variables have several different formats that are often confused with variable types. We'll see in a minute how variable view puts users on the wrong track here.
The only way to change a string variable to numeric or reversely is ALTER TYPE. However, there's several ways to make a numeric copy of a string variable or reversely. We'll get to those in a minute.

So What's Better: String or Numeric?

The simplest rule of thumb is that only nominal variables with many categories
should be string variables in SPSS. Examples are names of people, email addresses, passport numbers and so on. Although such variables can be useful, we don't usually analyze them.
We do sometimes analyze nominal variables with few categories -such as nationality, blood group or profession. If these are string variables, they may or may not cause trouble. For example, the independent variable for ANOVA may or may not be a string variable depending on the exact command you use for it.Precisely, UNIANOVA does and ONEWAY does not accept string variables as factors.
You may get away by leaving such variables as strings. However, copying them into numeric variables makes sure you'll avoid all trouble. A decent way to do so is AUTORECODE. For converting metric string variables -holding just numbers- into numeric variables, see SPSS Convert String to Numeric Variable.

Determining SPSS Variable Types

So how do we know if a variable is string or numeric? In SPSS versions 24 and higher, tiny icons in front of variable names tell us the variable type, format and even measurement level. The icon for “nominal” may contain a tiny “a” which indicates it's a string variable.

For SPSS versions 23 and earlier, we'll inspect our variable view and use the following rule:

• if Type says “String”, you're dealing with a string variable;
• if Type does not say “String”, you're dealing with a numeric variable.

SPSS suggests that “Date” and “Dollar” are variable types as well. However, these are formats, not types. The way they are shown here among the actual variable types (string and numeric) is one of SPSS’ most confusing features.

SPSS Variable Formats - Introduction

Let's now have a look at the data in data view as shown the screenshot below. We'll briefly describe the kinds of variables we see.

Regarding these data, we stated earlier that

is a string variable and
through are numeric variables and contain only numbers.

However, values such as “26-jan-2015” sure don't look like numbers, do they? This is because SPSS can display numbers in very different ways. These ways of displaying data values are referred to as variable formats.

Determining SPSS Variable Formats

As we saw earlier, “Type” under variable view shows a confusing mixture of variable types and formats. We'll see the actual formats by running display dictionary. Part of the result is shown by the screenshot below.

SPSS distinguishes print and write formats but we don't bother about this distinction. SPSS variable formats consist of two parts. One or more letters indicate the format family. Most of them speak to themselves, except for the first two variables:

• A (“Alphanumeric”) is the usual format for string variables;
• F, (“Fortran”) indicates a standard numeric variable.

Formats end with numbers, indicating the number of characters to be shown. If a period is present, the number after the period indicates the number of decimal places to be displayed. The figure below illustrates these points.

SPSS Common Variable Formats

The figure below now summarizes some common variable types and formats we'll encounter in SPSS.

Setting Variable Formats in SPSS

You can set variable formats for numeric variables with the FORMATS command. For example, formats weight (f4.3). shows weight with 3 decimal places. Doing so affects the output you create: most tables will add an extra decimal place for weight as well. If you'd like to see this for yourself, run the syntax below and compare the 2 resulting tables.

Keep in mind that changing variable formats does not change your data in any way. The actual values are still the exact same numbers. They are merely displayed differently.

Variable Types and Formats - Why Bother?

Basically, “what you see is not what you get” in data view. For example, we see $20.37 but the actual value is just 20.37. So we can identify products costing $20,- or more by running the syntax below: compute expensive = (price >= 20). We don't include the dollar sign in our syntax. Although SPSS shows a dollar sign in data view, the actual values are just numbers and these are what the syntax acts upon.

Or let's say we'd like to add 30 days to our date variable. We could do so by running compute newdate = datesum(date,30,'days'). The resulting values are 13644236937.72. These are the correct numbers but they'll display as readable dates only after running something like formats newdate (date11). Another reason for bothering about variable formats is setting decimals places for output tables. For SPSS version 22 onwards, OUTPUT MODIFY does the trick as shown below.

In a similar vein, CTABLES allows choosing different formats for different statistics in your output.

Final Notes

This tutorial was somewhat theoretical but it has a lot of practical consequences. I hope you found it helpful.

Thanks for reading!