diff --git a/src/linpred.jl b/src/linpred.jl
index 4274f575..01d2b918 100644
--- a/src/linpred.jl
+++ b/src/linpred.jl
@@ -157,11 +157,27 @@ function delbeta!(p::DensePredChol{T,<:Cholesky}, r::Vector{T}, wt::Vector{T}) w
 end
 
 function delbeta!(p::DensePredChol{T,<:CholeskyPivoted}, r::Vector{T}, wt::Vector{T}) where T<:BlasReal
-    cf = cholfactors(p.chol)
-    piv = p.chol.p
-    cf .= mul!(p.scratchm2, adjoint(LinearAlgebra.mul!(p.scratchm1, Diagonal(wt), p.X)), p.X)[piv, piv]
-    cholesky!(Hermitian(cf, Symbol(p.chol.uplo)))
-    ldiv!(p.chol, mul!(p.delbeta, transpose(p.scratchm1), r))
+    ch = p.chol
+    rnk = rank(ch)
+    if rnk == length(ch.p)
+        cf = cholfactors(ch)
+        piv = ch.p
+        cf .= mul!(p.scratchm2, adjoint(LinearAlgebra.mul!(p.scratchm1, Diagonal(wt), p.X)), p.X)[piv, piv]
+        cholesky!(Hermitian(cf, Symbol(ch.uplo)))
+        ldiv!(ch, mul!(p.delbeta, transpose(p.scratchm1), r))
+    else
+        xc = p.X
+        scr = mul!(p.scratchm1, Diagonal(wt), xc)
+        scr2 = mul!(p.scratchm2, adjoint(xc), scr)
+        ch2 = cholesky!(scr2, Val(true), tol = -one(T), check = false) 
+        delbeta = mul!(p.delbeta, adjoint(scr), r)
+        permute!(delbeta, ch.p)
+        for k=(rnk+1):length(delbeta)
+            delbeta[k] = -zero(T)
+        end
+        LAPACK.potrs!(ch2.uplo, view(ch2.factors, 1:rnk, 1:rnk), view(delbeta, 1:rnk))
+        invpermute!(delbeta, ch.p)
+    end
     p
 end
 
diff --git a/test/runtests.jl b/test/runtests.jl
index b54d9127..bdc97937 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -142,6 +142,24 @@ end
     @test isapprox(coef(m2p_dep_pos_kw), coef(m2p))
 end
 
+@testset "collinearity and weights" begin
+    rng = StableRNG(1234321)
+    x1 = randn(100)
+    x1_2 = 3 * x1
+    x2 = 10 * randn(100)
+    x2_2 = -2.4 * x2
+    y = 1 .+ randn() * x1 + randn() * x2 + 2 * randn(100)
+    df = DataFrame(y = y, x1 = x1, x2 = x1_2, x3 = x2, x4 = x2_2, weights = repeat([1, 0.5],50))
+    f = @formula(y ~ x1 + x2 + x3 + x4)
+    lm_model = lm(f, df, wts = df.weights)
+    X = [ones(length(y)) x1_2 x2_2]
+    W = Diagonal(df.weights)
+    coef_naive = (X'W*X)\X'W*y
+    @test lm_model.model.pp.chol isa CholeskyPivoted
+    @test rank(lm_model.model.pp.chol) == 3
+    @test filter(!=(0.0), coef(lm_model)) ≈ coef_naive atol=1e-6
+end
+
 @testset "saturated linear model" begin
     df = DataFrame(x=["a", "b", "c"], y=[1, 2, 3])
     model = lm(@formula(y ~ x), df)