What Is Covariance? How to Read Covariance in AMOS and SPSS

What Is Covariance

You are looking at AMOS output and see tables named Variances, Covariances, and Correlations. A covariance value can be positive, negative, or very close to 0, but the number itself does not tell you whether the relationship is strong or weak unless you also know the measurement units of the two variables. Understanding this point prevents you from concluding too quickly that a larger covariance means a stronger relationship between two variables.

Covariance is a measure that describes the direction in which two variables change together. For two observed variables X and Y, sample covariance is calculated as follows:

Cov(X,Y) = Σ[(Xi - X̄)(Yi - Ȳ)] / (n - 1)

Here, Xi and Yi are the values for each observation, X̄ and Ȳ are the means, and n is the number of observations. When X is above its mean and Y is also above its mean, the product of the deviations is usually positive. When one variable is above its mean while the other is below its mean, that product is usually negative.

A covariance of 0, or close to 0, indicates that the sample does not show a clear linear tendency for the variables to vary together according to this measure. Covariance is not a test of causality and does not by itself prove that X causes Y to change. It is one component of a covariance matrix, commonly used when estimating SEM models in AMOS or examining the correlation structure among variables.

The Meaning of Covariance in Quantitative Research

Covariance answers this question: when one variable deviates from its mean, how much does the other variable usually deviate in the same or the opposite direction, measured in the original units. The sign of covariance indicates the direction. A positive covariance shows that the two variables tend to increase or decrease together, while a negative covariance shows an opposite tendency.

For example, if perceived service quality and continuance intention are both measured on a 1 to 5 Likert scale, a positive covariance indicates that respondents with higher service-quality scores also tend to have higher intention scores. If one variable is waiting time and the other is satisfaction, a negative covariance may fit the hypothesis that people who wait longer tend to be less satisfied.

The size of covariance depends on the measurement scale. If you change income from millions of đồng to đồng, the covariance involving income also changes according to the conversion factor. You should therefore not take the covariance for X and Y and compare it directly with the covariance for A and B to claim that one pair has a stronger relationship.

In AMOS, covariance commonly appears between exogenous variables, between measurement errors, or in the model-estimation section. Read the location of the arrow and the table name carefully. An estimated covariance between two variables does not mean that the model has confirmed a causal relationship between them. If the model has a one-way arrow from X to Y, you are reading a regression weight or path coefficient, not covariance.

If your goal is to check whether unusual observations are affecting the covariance, see the guide on outliers. If your goal is to test whether one variable changes the effect of another, that is a moderator problem. See moderator or moderating, rather than only reading covariance. You can also review the Statistics topic when you need to cross-check related analysis steps.

What Covariance Value Is Acceptable

Covariance does not have a general cutoff such as “0.5 or above is acceptable.” The reason is that this measure depends on the units and the dispersion of the two variables. When reporting covariance, focus on the sign, the statistical significance if the software provides it, and its relationship with correlation or the research hypothesis.

The table below summarizes how related measures are used. The thresholds in the table apply only to the stated measure and must not be transferred directly to covariance.

If you are asking “what covariance value is acceptable,” the practical answer is that you cannot decide from one covariance number alone. You need to check whether the covariance has the expected direction, whether it differs from 0 according to the accompanying test, and whether the model has been specified correctly.

In CFA or SEM, covariance between two exogenous variables can be freely estimated when there is a theoretical basis for it. Adding covariance only because modification indices suggest it may make the model fit the sample data better, but it can also make the model harder to explain. When using AMOS, record the theoretical reason for every covariance you add and report the model change transparently.

How to Read Covariance in the Output

In AMOS, you commonly see the Covariances, Variances, and Correlations tables in the Estimates section. Column names may include Estimate, S.E., C.R., P, and sometimes Label. Estimate is the estimated covariance value. S.E. is the standard error. C.R. is commonly used to assess the ratio between the estimate and its standard error. P is the p-value for the parameter test.

The following table is illustrative output, not the result of a real study. The variable names are set up to resemble a shortened AMOS output.

The row SAT <--> TRUST means that the estimated covariance between SAT and TRUST is 0.284. The two-headed arrow <--> matters because it indicates a covariance relationship. You should not write that SAT affects TRUST based only on this row.

The P column may display *** in AMOS. Check the note for your software version, but in common reporting practice this symbol represents a very small p-value. When writing your thesis, report the exact value the software allows and combine it with the direction of the covariance. A positive covariance with a small p-value shows that the covariance parameter differs from 0 according to the test. It is not evidence of a causal relationship.

If you run SPSS, a Covariance Matrix table may appear in a procedure or an options dialog, depending on what you select. SPSS may also provide Correlations, Descriptive Statistics, and other statistics. Do not take a value from the Covariance Matrix and call it a correlation. Check the exact table name before copying it into Chapter 4.

How to write this in your thesis

You can report the result using this template: “[The results show that the covariance between [variable X] and [variable Y] is [value], with a [positive/negative] sign and a p-value of [value]. This result indicates that the two variables tend to vary [in the same/opposite direction] in the sample, but it should not be interpreted as a causal effect.” When replacing the values, use your own output and do not retain the illustrative numbers above.

What to Do When Covariance Does Not Meet Your Expectation

First, identify the criterion behind the phrase “does not meet the requirement.” Negative covariance is not automatically unacceptable. If the hypothesis predicts an inverse relationship, a negative sign may be appropriate. Conversely, a positive covariance is not enough to conclude that a hypothesis is supported if the accompanying test does not support it or if the measurement model has problems.

Second, check how the variables were coded. An observed variable written in a negative direction may need reverse coding before you calculate the scale score. If you skip this step, the covariance between that variable and variables measuring the same construct may change sign or become much smaller.

Third, check missing values, incorrectly entered values, and unusual coding groups. For example, some rows may use 99 for “no response,” but SPSS may read it as a real score. This distorts the mean, variance, and covariance. Review Variable View, the missing-value rules, and the frequency table before running the analysis again.

Fourth, check outliers using a defensible procedure. Do not delete a row simply because it makes the covariance look better. If you identify an unusual case, record the detection criterion, check the data source, and explain why you kept or removed it. Any model rerun after removing data must be reported clearly.

Fifth, review the theoretical model. If two variables are allowed to covary without a conceptual basis, adding the relationship only to reduce Chi-square is difficult to defend. If you are testing an effect relationship, use the regression path that matches the model instead of interpreting covariance as a path coefficient.

Finally, if the issue is with the scale, check the reliability and validity of the scale in the correct order. You can read what a scale is to distinguish observed variables, latent variables, and scale construction. Item removal should be based on both theoretical reasoning and statistical results, not on one covariance value alone.

Distinguishing Covariance from Variance and Correlation

Variance is the variance of one variable. It shows how the observations of that variable are dispersed around its mean. In AMOS output, the row SAT <--> SAT in the Variances table is the variance of SAT. Covariance is the covariance between two variables and usually appears in a row with two different variable names in the Covariances table.

Correlation is covariance standardized by the standard deviations of the two variables. The formula is rXY = Cov(X,Y) / (SDX × SDY). Because it is standardized, correlation is not directly dependent on measurement units and usually falls between -1 and 1. This is why correlation is more convenient when comparing the strength of association across pairs of variables.

Another common confusion is covariance versus correlation coefficient. Both measures describe the association between two variables, but correlation is easier to compare because it is standardized. If your research question asks you to assess the strength of a correlation, read the Correlations table or the appropriate correlation coefficient instead of substituting covariance.

Covariance is also different from a mediator and a moderator. A mediator explains the mechanism through which X affects Y through an intermediate variable, while a moderator changes the strength or direction of a relationship. See mediator if your output includes an indirect effect or direct effect. Covariance only describes a covariance relationship between variables in the estimated part of the model.

Common Errors

The first error is looking for a fixed covariance threshold and then labeling a result acceptable or unacceptable. This ignores measurement units, dispersion, and the model context. Switch to correlation when you need to compare the strength of relationships, while retaining covariance for reading the covariance matrix and estimation results.

The second error is confusing Variances with Covariances. A row with one variable repeated, such as TRUST <--> TRUST, is a variance. A row with two different variables, such as SAT <--> TRUST, is covariance. Check the table name and the <--> symbol before writing your interpretation.

The third error is writing “X affects Y” when the output contains only covariance. Causal language must be tied to a model with a one-way arrow and an appropriate coefficient. For covariance, use phrases such as “tend to vary in the same direction,” “tend to vary in opposite directions,” or “the covariance parameter differs from 0 according to the test.”

The fourth error is deleting data until covariance has the desired sign or p-value. This is result polishing and may be questioned during the defence. Every data change needs a predefined criterion, a processing log, and a defensible reason.

If you need to add covariance between measurement errors in AMOS, base the decision on the content of the observed variables and the theoretical rationale. Modification indices are only a technical suggestion. Report the adjusted model together with the reason for the change rather than only stating that the fit indices improved.

Frequently asked questions

What is covariance in AMOS?

Covariance in AMOS is the covariance between two variables or between two errors that the model allows to be related. A positive value indicates a tendency in the same direction, while a negative value indicates an opposite tendency. Read it in the Covariances table and do not confuse it with Variances or Regression Weights.

What covariance value is acceptable?

There is no general threshold that applies to every covariance. The value depends on the measurement scale and the units of the two variables. Check the sign, standard error, C.R., p-value, and theoretical basis of the modeled relationship.

Is negative covariance unacceptable?

No. Negative covariance may be appropriate when the hypothesis predicts that two variables will vary in opposite directions, such as waiting time and satisfaction. Conclude that it is inconsistent only when the sign or test result conflicts with the hypothesis, theory, and data coding.

How is covariance different from correlation?

Covariance retains the original measurement units, so its size is difficult to compare across pairs of variables. Correlation is covariance standardized by the standard deviations and usually ranges from -1 to 1, making it more convenient for interpreting the strength of an association.

Why is covariance in AMOS different from covariance in SPSS?

AMOS may estimate covariance within an SEM model, while SPSS may calculate sample covariance directly from the data or within a specific procedure. The values can differ because of missing-data handling, the analysis sample, the estimation method, or the model specification. Check that the observation list and coding procedure are the same before comparing them.

Should you add covariance based on modification indices?

Add it only when a theoretical basis and the measurement content explain the relationship. Modification indices do not replace model reasoning. After adding it, rerun the model and record the adjustment in the methods or results chapter.

Open your output again and mark each row in Covariances, Variances, and Correlations. Then record the sign, Estimate, S.E., C.R., and P separately before interpreting the result. If you need to run covariance and the related analyses on your own .sav or .csv file, use M4 data analysis from DoThesis.