This article describes the TREND function in Microsoft Office Excel 2003 and in later versions of Excel, illustrates how the function is used, and compares the results of the function in Excel 2003 and in later versions of Excel with results of TREND in earlier versions of Excel.
TREND is evaluated by calling the related function, LINEST. Extensive changes to LINEST in Excel 2003 and in later versions of Excel are summarized, and their implications for TREND are noted.
Microsoft Excel 2004 for Macintosh InformationThe statistical functions in Microsoft Excel 2004 for Macintosh were updated using the same algorithms as Excel 2003 and later versions of Excel. Any information in this article that describes how a function works or how a function was modified for Excel 2003 and for later versions of Excel also applies to Excel 2004 for Macintosh.
The TREND(known_y's, known_x's, new_x's, constant) function is used to perform Linear Regression. A least squares criterion is used and TREND tries to find the best fit under that criterion. Known_y's represent data on the "dependent variable" and known_x's represent data on one or more "independent variables". The TREND Help file discusses rare cases where the second or third argument may be omitted.
If the last argument "constant" is set to TRUE, you want the regression model to include a coefficient for the intercept in the regression model. If the last argument is set to FALSE, no intercept term is included; the fitted regression is forced to go through the origin. The last argument is optional; if omitted it is interpreted as TRUE.
For ease of exposition in the rest of this article, assume that data is arranged in columns so that known_y's is a column of y data and known_x's is one or more columns of x data. Of course the dimensions (lengths) of each of these columns must be equal. New_x's will also be assumed to be arranged in columns and there must be the same number of columns for new_x's as for known_x's. All the observations in this article are equally true if the data is not arranged in columns, but it is just easier to discuss this single (most frequently used) case.
After computing the best fit regression model (by essentially calling Excel's LINEST function), TREND returns predicted values associated with new_x's.
This article uses examples to show how TREND relates to LINEST and to point out problems with LINEST in Microsoft Excel 2002 and in earlier versions of Excel. These problems translate to problems with TREND. While the code for TREND was not rewritten for Excel 2003 and for later versions of Excel, extensive changes (and improvements) in LINEST code have been made.
TREND effectively calls LINEST, executes LINEST, uses regression coefficients in LINEST output in its calculation of predicted y values associated with each row of new_x's, and presents this column of predicted y values to you. Therefore, you must know about problems in the execution of LINEST.
As a supplement to this article, the following article about LINEST is highly recommended. It contains several examples and documents problems with LINEST in Excel 2002 and in earlier versions of Excel.
For additional information, click the following article number to view the article in the Microsoft Knowledge Base:
Description of the LINEST function in Excel 2003 and in Excel 2004 for Mac
Because the focus here is on numeric problems in Excel 2002 and in earlier versions of Excel, this article does not have many practical examples for how to use TREND. TREND's Help file contains useful examples.
TREND(known_y's, known_x's, new_x's, constant)
The arguments, known_y's, known_x's, and new_x's must be arrays or cell ranges with related dimensions. If known_y's is one column by m rows, then known_x's are c columns by m rows where c is greater than or equal to one. Note that c is the number of predictor variables; m is the number of data points. New_x's must then be c columns by r rows where r is greater than or equal to one. (Similar relationships in dimensions must hold if data is laid out in rows instead of columns.) Constant is a logical argument that must be set to TRUE or FALSE (or 0 or 1 that Excel interprets as FALSE or TRUE, respectively). The last three arguments to TREND are all optional; see the TREND Help file for options of omitting the second argument, the third argument, or both. Omitting the fourth argument is interpreted as TRUE.
The most common usage of TREND includes two ranges of cells that contain the data, such as TREND(A1:A100, B1:F100, B101:F108, TRUE). Note that because there is typically more than one predictor variable, the second argument in this example contains multiple columns. In this example, there are one hundred subjects, one dependent variable value (known_y) for each subject, and five dependent variable values (known_x's) for each subject. There are eight additional hypothetical subjects where you want to use TREND to compute predicted y values.
Example of usage
An Excel worksheet example is provided to illustrate the following key concepts:
- How TREND interacts with LINEST
- Problems that occur because of collinear known_x's in TREND (or LINEST) for Excel 2002 and for earlier versions of Excel
Extensive discussion of the second bullet point in the context of LINEST is provided in the article on LINEST.
To illustrate TREND collinearity, create a blank Excel worksheet, copy the following table, select cell A1 in your blank Excel worksheet, and then paste the entries so that the following table fills cells A1:K35 in your worksheet.
|TREND using cols B,C:||pre-Excel 2003 values:||Values in Excel 2003 and in later versions of Excel:|
|TREND using col B only|
|Fitted values from Excel 2003 and from later versions of Excel LINEST results|
|Using cols B, C||Using Col B|
|= K24*1 + J24*B7 + I24*C7||=J31*1+I31*B7|
|=K24*1 + J24*B8 + I24*C8||=J31*1 +I31*B8|
|LINEST using cols B,C:||pre-Excel 2003 values:||Values in Excel 2003 and in later versions of Excel:|
|LINEST using col B only|
After you paste this table into your new Excel worksheet, click Paste Options
, and then click Match Destination Formatting
. With the pasted range still selected, use one of the following procedures, as appropriate for the version of Excel that you are running:
- In Microsoft Office Excel 2007 and 2010, click the Home tab, click Format in the Cells group, and then click AutoFit Column Width.
- In Excel 2003, point to Column on the Format menu, and then click AutoFit Selection.
Data for TREND are in cells A1:C8. (Entries in cells D2:D6 are not part of the data, but are used for illustration later in this article.) Results of TREND for two different models for both earlier versions of Excel and later versions of Excel are presented in cells E10:E16 and cells I10:116, respectively. Results in cells A10:A16 will correspond to the version of Excel that you are using. For now, this article focuses on the results in Excel 2003 and in later versions of Excel when investigating how TREND calls LINEST and how TREND uses LINEST results.
TREND and LINEST can be viewed as interacting as follows:
- You call TREND(known_y's, known_x's, new_x's, constant).
- TREND calls LINEST(known_y's, known_x's, constant, TRUE).
- Regression coefficients from this call to LINEST are obtained; these coefficients appear in the first row of LINEST's output table.
- For each new_x's row, the predicted y-value is calculated based on these LINEST coefficients and the new_x's values in that row.
- The calculated value in step 4 is returned in the appropriate cell for TREND output corresponding to that new_x's row.
If TREND is to return appropriate results, then LINEST had better generate appropriate results in step 3. Problems here come from collinear predictor columns.
Predictor columns (known_x's) are collinear if at least one column, c, can be expressed as a sum of multiples of others, c1, c2, and other columns. Column c is frequently named redundant because the information that it contains can be constructed from the columns c1, c2, and other columns. The fundamental principle in the presence of collinearity is that results should be unaffected by including or removing a redundant column from the original data. Because LINEST in Excel 2002 and in earlier versions of Excel did not look for collinearity, this principle was easily violated. Predictor columns are almost collinear if at least one column, c, can be expressed as almost equal to a sum of multiples of others, c1, c2, and other columns. In this case "almost equal" means a very small sum of squared deviations of entries in c from corresponding entries in the weighted sum of c1, c2, and other columns; "very small" might be less than 10^(-12) for example.
The first model, in rows 10 to 12, uses columns B and C as predictors and requests Excel to model the constant (last argument set to TRUE). Excel then effectively inserts an additional predictor column that looks just like cells D2:D6. It is easy to notice that entries in column C in rows 2 to 6 are precisely equal to the sum of corresponding entries in columns B and D. Therefore, there is collinearity present because column C is a sum of multiples of:
- Column B
- Excel's additional column of 1s that is inserted because the third argument to LINEST (same as fourth argument to TREND) was omitted or TRUE (the "normal" case)
This causes such numeric problems that Excel 2002 and earlier versions of Excel cannot compute results and the TREND output table is filled with #NUM!.
The second model, in rows 14 to 16, is one that any version of Excel can handle successfully. There is no collinearity, and you can again request Excel to model the constant. This model is included here for two reasons.
First, it is perhaps most typical of practical cases: no collinearity present. These cases are handled well in all versions of Excel. It should be reassuring to know that numeric problems are not likely to occur in the most common practical case if you have an earlier version of Excel.
Second, this example is used to compare behavior of Excel 2003 and of later versions of Excel in the two models. Most major statistical packages analyze collinearity, remove a column that is a sum of multiples of others from the model, and alert you with a message such as "column C is linearly dependent on other predictor columns and has been removed from the analysis."
In Excel 2003 and in later versions of Excel, such a message is conveyed not in an alert or a text string, but in the LINEST output table. TREND has no mechanism for delivering such a message to you. In the LINEST output table, a regression coefficient that is zero and whose standard error is zero corresponds to a coefficient for a column that has been removed from the model. LINEST output tables are included in rows 23 to 35 corresponding to the TREND output in rows 10 to 16. The entries in cells I24:I25 show an eliminated redundant predictor column. In this case, LINEST chose to remove column C (coefficients in cells I24, J24, K24 correspond to columns C, B, and Excel's constant column, respectively). When there is collinearity present, any one of the columns involved can be removed and the choice is arbitrary.
In the second model in rows 30 to 35, there is no collinearity and no column removed. You can see that the predicted y values are the same in both models. This issue occurs because removing a redundant column that is a sum of multiples of others does not reduce the goodness of fit of the resulting model. Such columns are removed precisely because they represent no value added in trying to find the best least squares fit.
Also, if you examine LINEST output in Excel 2003 and in later versions of Excel in cells I23:K35, you will notice that the last three rows of the output tables are the same and that the entries in cells I31:J32 and cells J24:K25 coincide. This demonstrates that the same results are obtained when column C is included in the model but found to be redundant (output in cells I24:K28) as when column C was eliminated before LINEST was run (output in cells I31:J35). This satisfies the fundamental principle in the presence of collinearity.
In cells A18:C21, this article uses data for Excel 2003 and for later versions of Excel to illustrate how TREND takes LINEST output and computes the relevant predicted y-values. By examining the formulas in cells A20:A21 and cells C20:C21, you can see how LINEST coefficients are combined with new_x's data in cells B7:C8 for each of the two models (using columns B, C as predictors; using only column B as a predictor).
Collinearity is identified in LINEST in Excel 2003 and in later versions of Excel because of a completely different approach to solving for the regression coefficients. This approach is called QR Decomposition. The LINEST article describes a walkthrough of the QR Decomposition algorithm for a small example.
Summary of results in earlier versions of Excel
TREND results are adversely affected by inaccurate results in LINEST in Excel 2002 and in earlier versions of Excel.
LINEST was calculated by using an approach that paid no attention to collinearity issues. The existence of collinearity caused round-off errors, inappropriate standard errors of regression coefficients, and inappropriate degrees of freedom. Sometimes round-off problems were sufficiently severe that LINEST filled its output table with #NUM!.
If, as in the great majority of cases in practice, you can be confident that there are not collinear (or almost collinear) predictor columns, then LINEST generally provides acceptable results. Therefore, if you use TREND, you can be similarly reassured if you are confident that there are not collinear (or almost collinear) predictor columns.
Summary of results in Excel 2003 and in later versions of Excel
Improvements in LINEST include switching to the QR Decomposition method of determining regression coefficients. QR Decomposition has the following advantages:
- Better numeric stability (generally smaller round-off errors)
- Analysis of collinearity issues
All problems with Excel 2002 and with earlier versions of Excel that are illustrated in this article have been corrected for Excel 2003 and for later versions of Excel.
TREND's performance has been improved because LINEST has been greatly improved for Excel 2003 and for later versions of Excel. If you use an earlier version of Excel, verify that predictor columns are not collinear before using TREND.
Much of the material that is presented in this article and in the LINEST article might at first appear alarming to users of Excel 2002 and earlier versions of Excel. However, note that collinearity is a problem in a small percentage of cases. Earlier versions of Excel give acceptable TREND results when there is no collinearity.
Fortunately, improvements in LINEST also positively affect the Analysis ToolPak's linear regression tool (this calls LINEST) and two other related Excel functions: LOGEST and GROWTH.
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