# What is the difference between the two methods of lm on GLM.jl

**URL:** <https://discourse.julialang.org/t/what-is-the-difference-between-the-two-methods-of-lm-on-glm-jl/95140>\
**Category:** Statistics\
**Tags:** regression, glm\
**Created:** [February 24, 2023, 2:30pm UTC](https://discourse.julialang.org/t/what-is-the-difference-between-the-two-methods-of-lm-on-glm-jl/95140 "2023-02-24T14:30:02Z")\
**Posts on this page:** 3\
**Page:** 1

<div class="post-metadata">

**Author:** ![dpabon](https://avatars.discourse-cdn.com/v4/letter/d/da6949/32.png) [@dpabon](https://discourse.julialang.org/u/dpabon)\
**Post date:** [February 24, 2023, 2:30pm UTC](https://discourse.julialang.org/t/what-is-the-difference-between-the-two-methods-of-lm-on-glm-jl/95140/1 "2023-02-24T14:30:02Z")

</div>

The application of each one of the methods produced a different result:

```julia
using DataFrames
using GLM

new_x = rand(100,3)
new_y = rand(100)

GLM.lm(new_x, new_y)

```

Produce:

```julia
LinearModel{GLM.LmResp{Vector{Float64}}, GLM.DensePredChol{Float64, LinearAlgebra.CholeskyPivoted{Float64, Matrix{Float64}, Vector{Int64}}}}:

Coefficients:
───────────────────────────────────────────────────────────────
        Coef. Std. Error t Pr(>|t|) Lower 95% Upper 95%
───────────────────────────────────────────────────────────────
x1 0.295717 0.096847 3.05 0.0029 0.103502 0.487931
x2 0.462823 0.0882726 5.24 <1e-06 0.287627 0.63802
x3 0.0857036 0.0956069 0.90 0.3722 -0.10405 0.275457
───────────────────────────────────────────────────────────────

```

```julia
data_f = DataFrame(x1 = new_x[:,1], x2 = new_x[:,2], x3 = new_x[:,3], y = new_y)

ols = GLM.lm(@formula(y ~ x1 + x2), data_f)

```

Produce:

```julia
StatsModels.TableRegressionModel{LinearModel{GLM.LmResp{Vector{Float64}}, GLM.DensePredChol{Float64, LinearAlgebra.CholeskyPivoted{Float64, Matrix{Float64}, Vector{Int64}}}}, Matrix{Float64}}

y ~ 1 + x1 + x2

Coefficients:
──────────────────────────────────────────────────────────────────────────
                  Coef. Std. Error t Pr(>|t|) Lower 95% Upper 95%
──────────────────────────────────────────────────────────────────────────
(Intercept) 0.374513 0.0745926 5.02 <1e-05 0.226467 0.522558
x1 0.00540252 0.100866 0.05 0.9574 -0.194789 0.205594
x2 0.170571 0.0973294 1.75 0.0828 -0.0226013 0.363743
──────────────────────────────────────────────────────────────────────────

```

Why?

Thanks in advance for your help.

---

<div class="post-metadata">

**Author:** ![andreasnoack](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/andreasnoack/32/27_2.png) [@andreasnoack](https://discourse.julialang.org/u/andreasnoack)\
**Post date:** [February 24, 2023, 2:43pm UTC](https://discourse.julialang.org/t/what-is-the-difference-between-the-two-methods-of-lm-on-glm-jl/95140/2 "2023-02-24T14:43:24Z")

</div>

When using the formula, the intercept is automatically added. When passing a design matrix, it’s taken as given so you’d have to add a column on ones to match the formula behavior

```julia
julia> GLM.lm(@formula(y ~ x1 + x2 + x3), data_f)
StatsModels.TableRegressionModel{LinearModel{GLM.LmResp{Vector{Float64}}, GLM.DensePredChol{Float64, LinearAlgebra.CholeskyPivoted{Float64, Matrix{Float64}, Vector{Int64}}}}, Matrix{Float64}}

y ~ 1 + x1 + x2 + x3

Coefficients:
──────────────────────────────────────────────────────────────────────────
                 Coef. Std. Error t Pr(>|t|) Lower 95% Upper 95%
──────────────────────────────────────────────────────────────────────────
(Intercept) 0.472579 0.0959524 4.93 <1e-05 0.282115 0.663043
x1 -0.133307 0.106589 -1.25 0.2141 -0.344884 0.0782694
x2 0.086871 0.104626 0.83 0.4084 -0.120809 0.294551
x3 0.131423 0.101629 1.29 0.1991 -0.0703094 0.333155
──────────────────────────────────────────────────────────────────────────

julia> GLM.lm([ones(size(new_y, 1)) new_x], new_y)
LinearModel{GLM.LmResp{Vector{Float64}}, GLM.DensePredChol{Float64, LinearAlgebra.CholeskyPivoted{Float64, Matrix{Float64}, Vector{Int64}}}}:

Coefficients:
─────────────────────────────────────────────────────────────────
        Coef. Std. Error t Pr(>|t|) Lower 95% Upper 95%
─────────────────────────────────────────────────────────────────
x1 0.472579 0.0959524 4.93 <1e-05 0.282115 0.663043
x2 -0.133307 0.106589 -1.25 0.2141 -0.344884 0.0782694
x3 0.086871 0.104626 0.83 0.4084 -0.120809 0.294551
x4 0.131423 0.101629 1.29 0.1991 -0.0703094 0.333155
─────────────────────────────────────────────────────────────────

```

---

<div class="post-metadata">

**Author:** ![dpabon](https://avatars.discourse-cdn.com/v4/letter/d/da6949/32.png) [@dpabon](https://discourse.julialang.org/u/dpabon)\
**Post date:** [February 25, 2023, 3:00pm UTC](https://discourse.julialang.org/t/what-is-the-difference-between-the-two-methods-of-lm-on-glm-jl/95140/3 "2023-02-25T15:00:17Z")

</div>

Thanks for your help again, Andreas!!