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12. Dummy variables

Module items

R Script file code

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Lab assignment

Dummy variables

Sample lab assignment

Sample: Dummy variables

Suggested reading


  • Allen, Michael Patrick. 1997. “Regression Analysis with Dummy Variables.” Pp. 128–32 in Understanding Regression Analysis. Boston, MA: Springer.

Learning outcomes

  1. Explain how dummy variables represent categorical factor variables in linear regression
  2. Create dummy variables from binary and nominal variables and from groups based on continuous variables
  3. Determine which dummy variables to include in a regression model and identify the omitted comparison category
  4. Interpret coefficients of dummy variables in a linear regression model, referencing the omitted comparison category
  5. Compare the relative size of associations for dummy and continuous factor variables using standardized coefficients.

[[Dummy variable]] definition

  • Linear regression uses a continuous outcome variable. Factor variables can be continuous or categorical.
    • In the social sciences, we often work with categorical factor variables whose numbers are category codes rather than measurements.
  • A [[dummy variable]] is a variable coded zero (0) or one (1) to represent a category in a regression model.
    • Dummy variables allow us to compare categories. For example, we can compare the predicted personal income of females with that of males.

Examples

Example: Rent prices

  • Imagine a linear regression analysis in which we examine the association between population of the city (continuous) and house type (categorical: 1 for Townhouse, 2 for Studio) and the monthly rent (outcome variable).

  • We want to know whether predicted rent differs between townhouses and studios and how it is associated with the population of the city.

    • flowchart LR
subgraph F["Factor variables"]
    A[Population of city <br><br> Continuous variable <br><br> Min: 5K, Max: 10M]
end

subgraph H["Factor variables"]
    B[House type <br><br> Categorical variable <br><br> 1: Townhouse; 2: Studio]
end

subgraph O["Outcome variable"]
    C[Rent]
end

A -.->|Association?| C
B -.->|Association?| C

  • If we include the original house type variable in our model as it is, the linear regression model will incorrectly assume that:

    • Living in a "Studio" (2) is somehow twice the value of living in a “Townhouse” (1).
    • This would be similar to estimating the coefficients for age; 20-year-old and 40-year-old respondents (40 is twice 20).
  • The solution:
    • We use [[dummy variable]]s, which have only two values, zero (0) and one (1).
      • In the townhouse dummy variable, 1 means townhouse and 0 means not townhouse.

      • In the studio dummy variable, 1 means studio and 0 means not studio.

        • We create both variables to learn their coding, but a regression model with an intercept needs only one of these two dummy variables.

        • respondent Original variable Dummy variable 1 Dummy variable 2
          housetype townhouse studio
          1 2 (studio) 0 1
          2 1 (townhouse) 1 0
          3 2 (studio) 0 1
          4 1 (townhouse) 1 0
          5 2 (studio) 0 1
          6 1 (townhouse) 1 0

GSS example: Sex

  • The original sex variable uses 1=Male and 2=Female in GSS.
  • If we have a variable for sex with two responses (1=Male and 2=Female), we can't use the original values of 1 and 2, and interpret that as meaning being female is somehow two times of that being male.
    • The solution is to use dummy variables - variables with only two values, zero and one.
  • It does make sense to create a variable called “male" and interpret it as meaning that someone assigned a 1 on this variable is male and someone with a 0 is not.

    • Following this procedure, we also create a separate “female” dummy variable to show the effects of being female on the outcome variable, such as "personal income."

      • respondent Original variable Dummy variable 1 Dummy variable 2
        sex male female
        1 1 (male) 1 0
        2 1 (male) 1 0
        3 2 (female) 0 1
        4 1 (male) 1 0
        5 2 (female) 0 1
        6 2 (female) 0 1

Dummy variable coding structure

  • If a variable has two categories, we can create two dummy variables, one for each category.

    • Variable name Variable label Variable type Question wording and response categories
      sex Respondents' sex Binary What's your sex?

      (1: Male; 2: Female)

  • Since we create new variables using the dummy variable codes below, we also need to write these dummy variables' new variable labels. For example:

    • Respondents' sex (1: Male; 2: Female)
      • Being male
      • Being female
    • Respondents' immigrant status (1: Yes; 2: No)
      • Being nonimmigrant
      • Being immigrant
    • Belief in life after death (1: Yes; 2: No)
      • Believing in life after death
      • Not believing in life after death
    • Level of finding life exciting (1: Exciting; 2: Routine; 3: Dull)
      • Finding life exciting
      • Finding life routine
      • Finding life dull
    • Confidence level in education (1: A great deal; 2: Only some; 3: Hardly any)
      • Having a great deal of confidence in education
      • Having only some confidence in education
      • Having hardly any confidence in education
    • Home ownership status (1: Own; 2: Rent)
      • Owning a house
        • Or Having a house
      • Not owning a house
        • Or Not having a house

  • [[Model code]]

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      gss$dummyvariable1_name <- 
ifelse(gss$orig_var == value, 1, 0,
label = "Dummy variable's variable label")

gss$dummyvariable2_name <- 
ifelse(gss$orig_var == value, 1, 0,
label = "Dummy variable's variable label")

  • [[Working code]]

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      gss$male <- 
ifelse(gss$sex == 1, 1, 0,
label = "Being male")

gss$female <- 
ifelse(gss$sex == 2, 1, 0,
label = "Being female")
      • Line 1: We put the new variable name here:
        • male here ➜ dummyvariable1_name
          • It's better to write the value label here, thus male. Something simple and memorable.
      • Line 2: We put the original variable that we want to create a dummy variable.
        • sex here ➜ orig_var
        • since 1 is male in GSS dataset, we put 1 here ➜ value
          • The whole code with the last 1, 0 means:
            • "if sex is 1, create a new variable called “male”, assign them “1”, and assign the rest “0”".
      • Line 3: We write this new dummy variable's variable label here "Being male"
      • Line 5: We put the new variable name here:
        • female here ➜ dummyvariable2_name
          • It is useful to choose a simple and memorable variable name, such as female.
      • Line 6: We put the original variable that we want to create a dummy variable.
        • sex here ➜ orig_var
        • since 2 is female in GSS dataset, we put 2 here ➜ value
          • The whole code with the last 1, 0 means:
            • "if sex is 2, create a new variable called “female”, assign them “1”, and assign the rest “0”".
      • Line 7: We write this new dummy variable's variable label here "Being female"
      • Creating a dummy variable is in a way recoding a variable, and thus creating a new variable.

Adding dummy variable to a regression model

  • Let's add the dummy variables we have just created to the model we used in Linear regression basics module.

    • [[Model code]]

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        model5 <- lm(outcome_var ~ factor_var1 + factor_var2 + factor_var3 + factor_var4 + factor_var5 + factor_var6, data = gss)
tab_model(model5, show.std = T, show.ci = F, collapse.se = T)

    • [[Working code]]

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        model5 <- lm(conrinc ~ res16 + age + prestg10 + educ + male + female, data = gss)
tab_model(model5, show.std = T, show.ci = F, collapse.se = T)
        • Line 1: We put conrinc here ➜ outcome_here; res16 here ➜ factor_var1; age here ➜ factor_var2; prestg10 here ➜ factor_var3; educ here ➜ factor_var4; male here ➜ factor_var5; female here ➜ factor_var6.
          • Outcome variable variable first; then, factor variables separated by plus (+).
        • Line 2: Check the first argument: model5. If this is model5, then we should use model5 here
          • This needs to be model5, otherwise this code won't work.

  • Respondents' personal income

    • Factors Coeff. std. Coeff. p
      (Intercept) -48487.47
      (3955.96)      		 -0.00
      (0.02)      		 0.001***
      Population density of residence during
      adolescence years      		 570.97
      (409.78)      		 0.03
      (0.02)      		 0.164
      Respondents' age 191.36
      (40.68)      		 0.09
      (0.02)      		 0.001***
      Respondents' occupational prestige score 624.94
      (49.18)      		 0.26
      (0.02)      		 0.001***
      Respondents' education in years 2786.02
      (252.43)      		 0.23
      (0.02)      		 0.001***
      Being male 11787.50
      (1187.55)      		 0.18
      (0.02)      		 0.001***
      Observations 2268
      R² / R² adjusted 0.231 / 0.229
      • RStudio created the table, but where's female?

      • Actually, RStudio console part shows: "Parameters female were not estimable :

        • Model matrix is rank deficient. Parameters `female` were not
estimable.
Model matrix is rank deficient. Parameters `female` were not
estimable.

      • First, let's interpret the coefficient for the male dummy variable:

        • Being male is associated with $11,788 higher predicted personal income than being female.
          • We inherently know that “being female is associated with $11,788 lowe predicted personal income compared to being female.” Therefore, adding both dummy variables is redundant.
            • If we included female instead of male, its coefficient would be approximately -11,787.50, and the comparison group would be male.

[[Omitting dummy variable]]

  • Then, which dummy variable(s) to include and which one(s) to omit?
    • We include male if we want to compare males with females on predicted personal income.
    • We include female if we want to compare females with males on predicted personal income.
      • Either choice produces the same predicted values when the same respondents and other factors are used.
        • We include one dummy variable and omit the other as the comparison category.
          • If both are included, R cannot estimate both coefficients separately and it will remove the last added dummy variable.

  • Omitting a dummy variable does not mean removing respondents from that category:

    • We actually attribute a significant meaning to the omitted dummy variable that:

      • The omitted variable becomes our [[comparison dummy variable]]:

        • Its category remains in the data and is used as the reference for interpreting the included dummy variable.

          • Interpretation of dummy variables

            Being male (included dummy variable) is associated with $11,788 higher predicted personal income than being female (omitted - comparison dummy variable).

            OR

            Being female (included dummy variable) is associated with $11,788 lower predicted personal income than being male (omitted - comparison dummy variable).

    • Likewise,

      • if we create two dummy variables using a single original variable:
        • We need to include one, and omit one.
      • if we create three dummy variables using a single original variable:
        • We need to include two, and omit one.
      • if we create four dummy variables using a single original variable:
        • We need to include three, and omit one.

GSS example: Predicting personal income (conrinc)

Find the variables in Variables in GSS page

  • In this analysis, we examine whether the factor variables are associated with higher or lower personal income. Linear regression alone does not establish that they cause changes in personal income.

    • flowchart LR
subgraph C0[Numerical factor variables]
    direction TB
    A[Population density]
    B[Age]
    C[Occupational prestige]
    D[Education]
end

subgraph D0[Dummy factor variables]
    subgraph I0[Immigration status]
        direction TB
        I1[Being nonimmigrant]
        I2[Being immigrant]
    end

    subgraph M0[Marital status]
        direction TB
        M1[Being married]
        M2[Being formerly in union]
        M3[Being single ]
    end

    subgraph S0[Socio-economic status]
        direction TB
        S1[Low SES]
        S2[Moderate SES]
        S3[High SES]
    end
end

subgraph O0[Outcome variable]
    E[Personal income]
end

A -.->|Association?| E
B -.->|Association?| E
C -.->|Association?| E
D -.->|Association?| E
I0 -.->|Association?| E
M0 -.->|Association?| E
S0 -.->|Association?| E

  • We want to check the variable types: conrinc is continuous, res16 is a coded residence category, and age, prestg10, and educ are treated as numerical factors. We also need to identify the categories of the variables used to create dummy variables.

  • [[Search]] these variable names in Variables in GSS page.

    • Variable name Variable label Variable type Question wording and response categories
      conrinc Respondents' personal income Continuous What is your income in dollars?

      (Min: $281.5; Max, $123,761.9)
      res16 Population density of residence during adolescence years Ordinal ✅ Which of the categories on this card comes closest to the type of place you were living in when you were 16 years old?

      (1: Country, nonfarm; 2: Farm; 3: Town less than 50K; 4: 50K to 250K; 5: Big city, suburb; 6: City greater than 250K)
      age Respondents' age Continuous What is your age?

      (Min: 18, Max: 89)
      prestg10 Respondents' occupational prestige score Continuous Respondent's occupational prestige score (calculated)

      (Min: 16, Max: 80)
      educ Respondents' education in years Continuous What is the highest year of school you completed?

      (Min: 0, Max: 20)
      born Respondents' immigrant status Binary Were you born in this country?

      (1: Yes; 2: No)
      marital Respondents' marital status Nominal Are you currently — married, widowed, divorced, separated, or have you never been married?

      (1: Married; 2: Widowed; 3: Divorced; 4: Separated; 5: Never married)
      sei10 Respondents' socio-economic index score Continuous Socio-economic index score of the respondent (calculated)

      (Min: 9, Max: 92.8)

[[Dummy variable]]: Categorical (binary) #code

  • [[Model code]]

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      gss$dummyvar1 <- 
ifelse(gss$orig_var == value, 1, 0,
label = "Dummy variable's variable label")

gss$dummyvar2 <- 
ifelse(gss$orig_var == value, 1, 0,
label = "Dummy variable's variable label")

  • [[Working code]]

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      gss$nonimmigrant <- 
ifelse(gss$born == 1, 1, 0,
label = "Being nonimmigrant")

gss$immigrant <- 
ifelse(gss$born == 2, 1, 0,
label = "Being immigrant")
      • Line 1: We put the new variable name here:
        • nonimmigrant here ➜ dummyvar1
          • It's better to write something simple and memorable value label here, thus nonimmigrant.
      • Line 2: We put the original variable that we want to create a dummy variable.
        • born here ➜ orig_var
        • since 1 is "yes to being born in this country" in GSS dataset, we put 1 here ➜ value
          • The whole code with the last 1, 0 means:
            • "if born is 1, create a new variable called nonimmigrant, assign them “1”, and assign the rest “0”".
      • Line 3: We write this new dummy variable's variable label here "Being nonimmigrant"
      • Line 5: We put the new variable name here:
        • immigrant here ➜ dummyvar2
          • It's better to write something simple and memorable here, thus immigrant.
      • Line 6: We put the original variable that we want to create a dummy variable.
        • born here ➜ orig_var
        • since 2 is "no to being born in this country" in GSS dataset, we put 2 here ➜ value
          • The whole code with the last 1, 0 means:
            • "if born is 2, create a new variable called immigrant, assign them “1”, and assign the rest “0”".
      • Line 7: We write this new dummy variable's variable label here "Being immigrant"
      • Creating a dummy variable is in a way recoding a variable, and thus creating a new variable.
        • [[Find this working code in the R script file]].
          • [[Highlighting and running]] this code will create two more variables, nonimmigrant and immigrant.

[[Dummy variable]]: Categorical (nominal/ordinal) #code

  • [[Model code]]

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      gss$dummyvar1 <- 
ifelse(gss$orig_var == value, 1, 0,
label = "Dummy variable's variable label")

gss$dummyvar2 <- 
ifelse(gss$orig_var == value | gss$orig_var == value | gss$orig_var == value, 1, 0,
label = "Dummy variable's variable label")

gss$dummyvar3 <- 
ifelse(gss$orig_var == value, 1, 0,
label = "Dummy variable's variable label")

  • [[Working code]]

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      gss$married <- 
ifelse(gss$marital == 1, 1, 0,
label = "Being married")

gss$formerlyunion <- 
ifelse(gss$marital == 2 | gss$marital == 3 | gss$marital == 4, 1, 0,
label = "Being formerly in union")

gss$single <- 
ifelse(gss$marital == 5, 1, 0,
label = "Being single")
      • These codes follow the same approach as the dummy variable codes for binary variables.
        • The exception is the second dummy variable in which we merge some categories.
      • Line 5: We put the new variable name here:
        • formerlyunion here ➜ dummyvar2
          • It's better to write something simple and memorable, thus formerlyunion.
      • Line 6: We put the original variable that we want to create a dummy variable.
        • marital here ➜ orig_var
        • Since we want to merge respondents who are 2 "Widowed", 3 "Divorced", and 4 "Separated", we put marital == 2 | gss$marital == 3 | gss$marital == 4, 1, 0 here ➜ gss$orig_var == value | gss$orig_var == value | gss$orig_var == value, 1, 0
          • The whole code with the last 1, 0 means:
            • "if marital is 2 or 3 or 4, create a new variable called formerlyunion, assign them “1”, and assign the rest “0”".
      • Line 7: We write this new dummy variable's variable label here "Being formerly in union"
        • Creating a dummy variable is in a way recoding a variable, and thus creating a new variable.
          • [[Find this working code in the R script file]].
            • [[Highlighting and running]] this code will create three more variables, married, formerlyunion, and single.

[[Dummy variable]]: Continuous #code

  • [[Model code]]

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      gss$dummyvar1 <- 
ifelse(gss$orig_var <= value, 1, 0,
label = "Dummy variable's variable label")

gss$dummyvar2 <- 
ifelse(gss$orig_var >= lowest_value & gss$orig_var <= highest_value, 1, 0,
label = "Dummy variable's variable label")

gss$dummyvar3 <- 
ifelse(gss$orig_var >= value, 1, 0,
label = "Dummy variable's variable label")

  • [[Working code]]

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      gss$lowses <- 
ifelse(gss$sei10 <= 40, 1, 0,
label = "Having low socio-economic status")

gss$moderateses <- 
ifelse(gss$sei10 >= 41 & gss$sei10 <= 75, 1, 0,
label = "Having moderate socio-economic status")

gss$highses <- 
ifelse(gss$sei10 >=76, 1, 0,
label = "Having high socio-economic status")
      • Line 1: We put the new variable name here:
        • lowses here ➜ dummyvar1
          • It's better to write something simple and memorable value label here, thus lowses.
      • Line 2: We put the original variable that we want to create a dummy variable.
        • sei10 here ➜ orig_var
        • since we define low ses as "40 and all values below 40", we put 40 here ➜ value
          • The whole code with the last 1, 0 means:
            • "if sei10 is 40 and below, create a new variable called lowses, assign them “1”, and assign the rest “0”".
      • Line 3: We write this new dummy variable's variable label here "Having low socio-economic status"
      • Line 5: We put the new variable name here:
        • moderateses here ➜ dummyvar2
          • It's better to write something simple and memorable value label here, thus moderateses.
      • Line 6: We put the original variable that we want to create a dummy variable.
        • sei10 here ➜ orig_var
        • We define moderate SES as scores from 41 to 75, including both endpoints. We put 41 here ➜ lowest_value and 75 here ➜ highest_value.
          • The whole code with the last 1, 0 means:
            • "if sei10 is 41 or higher and 75 or lower, create a new variable called moderateses, assign them “1”, and assign the rest “0”".
      • Line 7: We write this new dummy variable's variable label here "Having moderate socio-economic status"
      • Line 9: We put the new variable name here:
        • highses here ➜ dummyvar3
          • It's better to write something simple and memorable value label here, thus highses.
      • Line 10: We put the original variable that we want to create a dummy variable.
        • sei10 here ➜ orig_var
        • since we define high ses as "76 or higher" we put 76 here ➜ value
          • The whole code with the last 1, 0 means:
            • "if sei10 is 76 and higher, create a new variable called highses, assign them “1”, and assign the rest “0”".

      • Line 11: We write this new dummy variable's variable label here "Having high socio-economic status"

        • Creating a dummy variable is in a way recoding a variable, and thus creating a new variable.
          • [[Find this working code in the R script file]].
            • [[Highlighting and running]] this code will create three more variables, lowses, moderateses, and highses.

      • Clarification on codes

        • Creating dummy variables using continuous variables requires a different coding structure.
          • This is because continuous variables have many possible values.
          • sei10 is the socio-economic index score of the respondent with minimum value of 9 and maximum value of 92.8.
            • We will create three separate categories for this continuous variable:
              • 1: Less than or equal to 40
                • This means respondents with a sei10 score of 40 or lower.
                • Code:
                  • ifelse(gss$sei10 <= 40, 1, 0)
              • 2: From 41 to 75
                • This means respondents with a sei10 score from 41 through 75, including decimals within this range.
                • Code:
                  • ifelse(gss$sei10 >= 41 & gss$sei10 <= 75, 1, 0)
              • 3: Greater than or equal to 76
                • This means respondents with a sei10 score of 76 or higher.
                • Code:
                  • ifelse(gss$sei10 >= 76, 1, 0)
              • In these codes:
                • <= means less than or equal to.
                  • Example: sei10 <= 40
                  • This includes 40 and all values below 40.
                • >= means greater than or equal to.
                  • Example: sei10 >= 76
                  • This includes 76 and all values above 76.
                • >= and <= include the boundary values in the moderate SES category.
                  • For example, a score of 41 belongs to moderate SES.
                  • For example, a score of 74 belongs to moderate SES.
                • & means and.
                  • We use & when both conditions must be true at the same time.
                  • This means:
                    • The sei10 score must be 41 or higher
                      • and
                    • The sei10 score must be 75 or lower

[[Linear regression]] with [[dummy variable]]s #code

  • [[Model code]]

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      model5 <- lm(outcome_here ~ factor1_here + factor2_here + factor3_here + factor4_here + factor5_here + factor6_here + factor7_here + factor8_here + factor9_here + factor10_here, data = gss)
tab_model(model5, show.std = T, show.ci = F, collapse.se = T)

  • [[Working code]]

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      model5 <- lm(conrinc ~ res16 + age + prestg10 + educ + male + immigrant + married + single + lowses + moderateses, data = gss)
tab_model(model5, show.std = T, show.ci = F, collapse.se = T)
      • Line 1: We put conrinc here ➜ outcome_here; res16 here ➜ factor1_here; age here ➜ factor2_here; prestg10 here ➜ factor3_here; educ here ➜ factor4_here; male here ➜ factor5_here; immigrant here ➜ factor6_here; married here ➜ factor7_here; single here ➜ factor8_here; lowses here ➜ factor9_here; moderateses here ➜ factor10_here.
        • Outcome variable variable first; then, factor variables separated by plus (+).
      • Line 2: Check the first argument: model5. If this is model5, then we should use model5 here.
        • This needs to be model5, otherwise this code won't work.
          • [[Find this working code in the R script file]].
            • [[Highlighting and running]] this code will generate the output below (which will appear in the [[viewer tab]] of RStudio).

[[Linear regression]] with [[dummy variable]]s #output

  • Respondents' personal income

    • Factors Coeff. std. Coeff. p
      (Intercept) 1716.51
      (6516.85)      		 -0.00
      (0.02)      		 0.792
      Population density of residence during
      adolescence years      		 432.41
      (400.58)      		 0.02
      (0.02)      		 0.281
      Respondents' age 67.87
      (44.24)      		 0.03
      (0.02)      		 0.125
      Respondents' occupational prestige score 154.14
      (68.21)      		 0.07
      (0.03)      		 0.024*
      Respondents' education in years 1983.68
      (253.42)      		 0.16
      (0.02)      		 < 0.001***
      Being male 10472.43
      (1154.80)      		 0.16
      (0.02)      		 < 0.001***
      Being immigrant 2133.64
      (1610.78)      		 0.02
      (0.02)      		 0.185
      Being married 6191.99
      (1583.73)      		 0.10
      (0.02)      		 < 0.001***
      Being single -5568.24
      (1765.27)      		 -0.08
      (0.03)      		 0.002**
      Having low socio-economic status -21005.16
      (2579.48)      		 -0.33
      (0.04)      		 < 0.001***
      Having moderate socio-economic status -5320.48
      (1974.35)      		 -0.08
      (0.03)      		 0.007**
      Observations 2260
      R² / R² adjusted 0.290 / 0.287

[[Linear regression]] with [[dummy variable]]s #interpretation

  • Linear regression interpretation sample

    • First section: The significance levels
      • Respondents' occupational prestige score, respondents' education in years, being male, being married, being single, having low socio-economic status, and having moderate socio-economic status have statistically significant associations with respondents' personal income since their p-values are less than 0.05.
      • Population density of residence during adolescence years, respondents' age, and being immigrant are not statistically significant factors of respondents' personal income since their p-values are greater than 0.05.
    • Second section: The explanation of coefficients
      • One additional point of respondents' occupational prestige score is associated with $154 higher predicted personal income.
      • One additional year of respondents' education is associated with $1,984 higher predicted personal income.
      • Being male is associated with $10,472 higher predicted personal income than being female.
      • Being married is associated with $6,192 higher predicted personal income than being formerly in union.
      • Being single is associated with $5,568 lower predicted personal income than being formerly in union.
      • Having low socio-economic status is associated with $21,005 lower predicted personal income than having high socio-economic status.
      • Having moderate socio-economic status is associated with $5,320 lower predicted personal income than having high socio-economic status.
    • Third section: The explanation of standardized coefficients
      • The largest standardized association with respondents' personal income is for having low socio-economic status (std. Coeff=-0.33), followed by respondents' education in years and being male (both std. Coeff=0.16), being married (std. Coeff=0.10), being single and having moderate socio-economic status (both std. Coeff=-0.08), and respondents' occupational prestige score (std. Coeff=0.07).
    • Fourth section: The explanation of adjusted R-squared
      • The adjusted R-squared value indicates that the model, which includes population density of residence during adolescence years, age, occupational prestige score, education in years, sex, immigrant status, marital status, and socio-economic status, accounts for 28.7% of the variation in respondents' personal income.

  • Linear regression with dummy variables interpretation template

    • First section: The significance levels
      • [[Variable label]] of significant [[factor variable]] 1, variable label of significant factor variable 2, variable label of significant factor variable 3... have statistically significant associations with variable label of [[outcome variable]] since their p-values are less than 0.05.
      • [If any]: Variable label of nonsignificant factor variable 4, variable label of nonsignificant factor variable 5... are not statistically significant factors of variable label of outcome variable since their p-values are greater than 0.05.
    • Second section: The explanation of coefficients
      • One additional [year/point/unit/score/dollar of factor variable 1] of variable label of significant factor variable 1 is associated with [coefficient with outcome variable unit] higher/lower predicted variable label of outcome variable.
      • Variable label of included dummy variable 1 is associated with [coefficient with outcome variable unit] higher/lower predicted variable label of outcome variable than variable label of omitted - comparison dummy variable 1.
      • Variable label of included dummy variable 2 is associated with [coefficient with outcome variable unit] higher/lower predicted variable label of outcome variable than variable label of omitted - comparison dummy variable 2.
      • Variable label of included dummy variable 3 is associated with [coefficient with outcome variable unit] higher/lower predicted variable label of outcome variable than variable label of omitted - comparison dummy variable 3.
    • Third section: The explanation of standardized coefficients
      • The largest standardized association with variable label of outcome variable is for variable label of the strongest factor variable (std. Coeff=0.xx), followed by variable label of the second strongest factor variable (std. Coeff=0.xx), and variable label of the third strongest factor variable (std. Coeff=0.xx)...
    • Fourth section: The explanation of adjusted R-squared
      • The [[adjusted R-squared]] value indicates that the model, which includes variable label of factor variable 1, variable label of factor variable 2, variable label of factor variable 3..., accounts for [adjusted R-squared as a percentage] of the variation in variable label of outcome variable.

  • Interpretation explanation

    • First section: The significance levels
      • We identify the variable labels of the factor variables that have statistically significant associations with the outcome variable.
        • When the p-value is less than 0.05, we say the factor variable has a statistically significant association with the outcome variable.
        • When the p-value is greater than 0.05, we say the factor variable is not a statistically significant factor of the outcome variable.
    • Second section: The explanation of coefficients
      • We use the Coeff. column to explain how one-unit increases in statistically significant continuous or numerical factor variables are associated with higher or lower predicted values of the outcome variable.
        • We include the units of both the factor variable and outcome variable (year, point, unit, score, dollar, etc.).
      • For significant dummy variables, we explain how the included category differs in predicted outcome from the omitted comparison category.
        • Note: We do not include nonsignificant factor variables in this section.
    • Third section: The explanation of standardized coefficients
      • We use the std. Coeff. column to identify the factor variables with the largest standardized associations with the outcome variable.
        • We list statistically significant factor variables from the largest to the smallest absolute standardized coefficient. A negative coefficient can have a larger absolute value than a positive coefficient.
        • Note: We do not include nonsignificant factor variables in this section.
    • Fourth section: The explanation of adjusted R-squared
      • We report the adjusted R-squared as a percentage to explain how much of the variation in the outcome variable is accounted for by the model.
        • We list all factor variables included in the model, including those that are not statistically significant.